%------------------------------------------------------------------------------
% File     : ITP213_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Hoare_Triple 00154_004690
% 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    : 0025_Hoare_Triple_00154_004690 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12089 (2870 unt;1954 typ;   0 def)
%            Number of atoms       : 27617 (8162 equ)
%            Maximal formula atoms :   38 (   2 avg)
%            Number of connectives : 19968 (2486   ~; 336   |;1937   &)
%                                         (2162 <=>;13047  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   6 avg)
%            Maximal term depth    :   40 (   2 avg)
%            Number of types       :   13 (  12 usr)
%            Number of type conns  : 1735 (1410   >; 325   *;   0   +;   0  <<)
%            Number of predicates  :  232 ( 229 usr;   2 prp; 0-6 aty)
%            Number of functors    : 1713 (1713 usr;  66 con; 0-7 aty)
%            Number of variables   : 39068 (35209   !; 722   ?;39068   :)
%                                         (3137  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 15:20:36.002
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
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_Sum__Type_Osum,type,
    sum_sum: ( $tType * $tType ) > $tType ).

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

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

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

tff(ty_t_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_HOL_Obool,type,
    bool: $tType ).

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

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

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

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

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

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

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

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

% Explicit typings (1929)
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_Groups_Ominus,type,
    minus: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_Rings_Oring__1__no__zero__divisors,type,
    ring_15535105094025558882visors: 
      !>[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_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_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__boolean__algebra,type,
    comple489889107523837845lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
    comple592849572758109894attice: 
      !>[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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__hg____,type,
    aTP_Lamp_hg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hh____,type,
    aTP_Lamp_hh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hi____,type,
    aTP_Lamp_hi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hj____,type,
    aTP_Lamp_hj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hk____,type,
    aTP_Lamp_hk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hl____,type,
    aTP_Lamp_hl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hm____,type,
    aTP_Lamp_hm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hn____,type,
    aTP_Lamp_hn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ho____,type,
    aTP_Lamp_ho: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hp____,type,
    aTP_Lamp_hp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hq____,type,
    aTP_Lamp_hq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hr____,type,
    aTP_Lamp_hr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hs____,type,
    aTP_Lamp_hs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ht____,type,
    aTP_Lamp_ht: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__hv____,type,
    aTP_Lamp_hv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hw____,type,
    aTP_Lamp_hw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hx____,type,
    aTP_Lamp_hx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hy____,type,
    aTP_Lamp_hy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hz____,type,
    aTP_Lamp_hz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ia____,type,
    aTP_Lamp_ia: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ik____,type,
    aTP_Lamp_ik: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__il____,type,
    aTP_Lamp_il: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__im____,type,
    aTP_Lamp_im: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

tff(sy_c_ATP_058Lamp__is____,type,
    aTP_Lamp_is: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__it____,type,
    aTP_Lamp_it: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iu____,type,
    aTP_Lamp_iu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iv____,type,
    aTP_Lamp_iv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iw____,type,
    aTP_Lamp_iw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ix____,type,
    aTP_Lamp_ix: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iy____,type,
    aTP_Lamp_iy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

tff(sy_c_ATP_058Lamp__jh____,type,
    aTP_Lamp_jh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__jk____,type,
    aTP_Lamp_jk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jl____,type,
    aTP_Lamp_jl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jm____,type,
    aTP_Lamp_jm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jn____,type,
    aTP_Lamp_jn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jo____,type,
    aTP_Lamp_jo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jp____,type,
    aTP_Lamp_jp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jq____,type,
    aTP_Lamp_jq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__tj____,type,
    aTP_Lamp_tj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tk____,type,
    aTP_Lamp_tk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tl____,type,
    aTP_Lamp_tl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tm____,type,
    aTP_Lamp_tm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tn____,type,
    aTP_Lamp_tn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__to____,type,
    aTP_Lamp_to: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tp____,type,
    aTP_Lamp_tp: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uc____,type,
    aTP_Lamp_uc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ud____,type,
    aTP_Lamp_ud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ue____,type,
    aTP_Lamp_ue: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uf____,type,
    aTP_Lamp_uf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ug____,type,
    aTP_Lamp_ug: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ul____,type,
    aTP_Lamp_ul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__us____,type,
    aTP_Lamp_us: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ut____,type,
    aTP_Lamp_ut: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vm____,type,
    aTP_Lamp_vm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vn____,type,
    aTP_Lamp_vn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vo____,type,
    aTP_Lamp_vo: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wf____,type,
    aTP_Lamp_wf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wg____,type,
    aTP_Lamp_wg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wh____,type,
    aTP_Lamp_wh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wi____,type,
    aTP_Lamp_wi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wj____,type,
    aTP_Lamp_wj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wk____,type,
    aTP_Lamp_wk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wl____,type,
    aTP_Lamp_wl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wm____,type,
    aTP_Lamp_wm: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wu____,type,
    aTP_Lamp_wu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xi____,type,
    aTP_Lamp_xi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xj____,type,
    aTP_Lamp_xj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xl____,type,
    aTP_Lamp_xl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xn____,type,
    aTP_Lamp_xn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xp____,type,
    aTP_Lamp_xp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xq____,type,
    aTP_Lamp_xq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xr____,type,
    aTP_Lamp_xr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xs____,type,
    aTP_Lamp_xs: 
      !>[A: $tType,B: $tType] : 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] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Assertions_Oassn_OAbs__assn,type,
    abs_assn: fun(product_prod(heap_ext(product_unit),set(nat)),bool) > assn ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

tff(sy_c_Assertions_Oentails,type,
    entails: ( 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)),bool) ).

tff(sy_c_Assertions_Oin__range__rel,type,
    in_range_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).

tff(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

tff(sy_c_Assertions_Oone__assn__raw__rel,type,
    one_assn_raw_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).

tff(sy_c_Assertions_Oprecise,type,
    precise: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,assn)) > $o ) ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: bool > assn ).

tff(sy_c_Assertions_Opure__assn__raw,type,
    pure_assn_raw: 
      !>[A: $tType,B: $tType] : ( bool > fun(product_prod(A,set(B)),bool) ) ).

tff(sy_c_Assertions_Opure__assn__raw__rel,type,
    pure_assn_raw_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(bool,product_prod(A,set(B))),fun(product_prod(bool,product_prod(A,set(B))),bool)) ).

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)),bool) ) ).

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)),bool) ) ).

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)))),bool)) ).

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

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)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).

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)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

tff(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).

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_Octwo,type,
    bNF_Cardinal_ctwo: set(product_prod(bool,bool)) ).

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__Def_OGr,type,
    bNF_Gr: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Def_OGrp,type,
    bNF_Grp: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,bool)) ) ).

tff(sy_c_BNF__Def_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,bool) > fun(A,fun(A,bool)) ) ).

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).

tff(sy_c_BNF__Def_Orel__set,type,
    bNF_rel_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(set(A),fun(set(B),bool)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
    bNF_Greatest_Succ: 
      !>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
    bNF_Gr5436034075474128252omCard: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * B ) > A ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
    bNF_Greatest_image2: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_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_Ocurr,type,
    bNF_Wellorder_curr: 
      !>[A: $tType,B: $tType,C: $tType] : ( set(A) > fun(fun(product_prod(A,B),C),fun(A,fun(B,C))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
    bNF_We2720479622203943262_image: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
    bNF_We413866401316099525erIncl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
    bNF_Wellorder_ordIso: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
    bNF_Wellorder_ordLeq: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
    bNF_We4044943003108391690rdLess: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__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) ) > fun(A,bool) ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
    bNF_We1388413361240627857o_max2: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
    bNF_We6954850376910717587_minim: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
    bNF_Wellorder_wo_suc: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

tff(sy_c_Basic__BNFs_Opred__fun,type,
    basic_pred_fun: 
      !>[A: $tType,B: $tType] : ( fun(A,bool) > fun(fun(B,bool),fun(fun(A,B),bool)) ) ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : ( ( A * 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 > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,bool) ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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,bool) ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: fun(code_integer,int) ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun(int,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: fun(code_natural,nat) ).

tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: nat > code_natural ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
    complete_Sup_Sup: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
    comple1908693960933563346ssible: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,bool) ) > $o ) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
    comple7512665784863727008ratesp: 
      !>[A: $tType] : ( fun(A,A) > fun(A,bool) ) ).

tff(sy_c_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
    condit8047198070973881523_above: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
    condit8119078960628432327_below: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * 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_Oproj,type,
    equiv_proj: 
      !>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(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_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofilter_OAbs__filter,type,
    abs_filter: 
      !>[A: $tType] : ( fun(fun(A,bool),bool) > filter(A) ) ).

tff(sy_c_Filter_Ofiltercomap,type,
    filtercomap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofinite__subsets__at__top,type,
    finite5375528669736107172at_top: 
      !>[A: $tType] : ( set(A) > filter(set(A)) ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Filter_Oprincipal,type,
    principal: 
      !>[A: $tType] : ( set(A) > filter(A) ) ).

tff(sy_c_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : fun(set(B),nat) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
    finite6289374366891150609ommute: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),bool) ).

tff(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).

tff(sy_c_Finite__Set_Ofold__graph,type,
    finite_fold_graph: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > fun(B,bool) ) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

tff(sy_c_Fun_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).

tff(sy_c_Fun_Ooverride__on,type,
    override_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).

tff(sy_c_Fun_Ostrict__mono__on,type,
    strict_mono_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * 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_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
    semiring_gcd_Gcd_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

tff(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : 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(C,A) * set(C) ) > A ) ).

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__mult__class_Oprod__list,type,
    groups5270119922927024881d_list: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_HOL_Odefault__class_Odefault,type,
    default_default: 
      !>[A: $tType] : A ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Heap_Oaddr__of__ref,type,
    addr_of_ref: 
      !>[A: $tType] : ( ref(A) > nat ) ).

tff(sy_c_Heap_Oheap_Olim,type,
    lim: 
      !>[Z: $tType] : ( heap_ext(Z) > nat ) ).

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_OHeap__lub,type,
    heap_Time_Heap_lub: 
      !>[A: $tType] : ( set(heap_Time_Heap(A)) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oassert,type,
    heap_Time_assert: 
      !>[A: $tType] : ( ( fun(A,bool) * 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),bool) * 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_Olift,type,
    heap_Time_lift: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(A,heap_Time_Heap(B)) ) ).

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] : fun(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)))),bool)) ).

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_If,type,
    if: 
      !>[A: $tType] : ( ( bool * A * A ) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
    complete_lattice_gfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
    complete_lattice_lfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( ( set(A) * nat ) > A ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_Int_Onat,type,
    nat2: fun(int,nat) ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,bool)) ).

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
    lattices_ord_arg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
    lattic8678736583308907530nf_fin: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
    lattic4895041142388067077er_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
    lattic4630905495605216202up_fin: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lifting_Orel__pred__comp,type,
    rel_pred_comp: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,bool) * A ) > $o ) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).

tff(sy_c_List_Oarg__min__list,type,
    arg_min_list: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),bool)) ).

tff(sy_c_List_Obind,type,
    bind: 
      !>[A: $tType,B: $tType] : ( ( list(A) * fun(A,list(B)) ) > list(B) ) ).

tff(sy_c_List_Obutlast,type,
    butlast: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oconcat,type,
    concat: 
      !>[A: $tType] : ( list(list(A)) > list(A) ) ).

tff(sy_c_List_Ocoset,type,
    coset: 
      !>[A: $tType] : ( list(A) > set(A) ) ).

tff(sy_c_List_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( list(A) > fun(A,nat) ) ).

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OdropWhile,type,
    dropWhile: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( fun(A,bool) > fun(list(A),list(A)) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofold,type,
    fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * set(B) * fun(B,A) ) > $o ) ).

tff(sy_c_List_Ofoldl,type,
    foldl: 
      !>[B: $tType,A: $tType] : ( fun(B,fun(A,B)) > fun(B,fun(list(A),B)) ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(nat,set(product_prod(list(A),list(A)))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexordp,type,
    lexordp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * set(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
    linord329482645794927042rt_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__key,type,
    linorder_insort_key: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(B,fun(list(B),list(B))) ) ).

tff(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(list(B),list(B)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
    linord144544945434240204of_set: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),list(B)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : fun(set(A),list(A)) ).

tff(sy_c_List_Olinorder__class_Ostable__sort__key,type,
    linord3483353639454293061rt_key: 
      !>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(list(A),list(Aa)) ) ).

tff(sy_c_List_Olist_Orec__list,type,
    rec_list: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,fun(list(A),fun(C,C))) ) > fun(list(A),C) ) ).

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : fun(list(A),set(A)) ).

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).

tff(sy_c_List_Olist_Otl,type,
    tl: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Olist__update,type,
    list_update: 
      !>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).

tff(sy_c_List_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olistrel1p,type,
    listrel1p: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

tff(sy_c_List_Olists,type,
    lists: 
      !>[A: $tType] : ( set(A) > set(list(A)) ) ).

tff(sy_c_List_Olistset,type,
    listset: 
      !>[A: $tType] : ( list(set(A)) > set(list(A)) ) ).

tff(sy_c_List_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).

tff(sy_c_List_Omap__project,type,
    map_project: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > set(B) ) ).

tff(sy_c_List_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))),bool)) ).

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),bool)) ).

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oord_Olexordp,type,
    lexordp2: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),fun(list(A),bool)) ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : ( fun(A,bool) > fun(list(A),product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oproduct__lists,type,
    product_lists: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Oremdups,type,
    remdups: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Oremdups__adj,type,
    remdups_adj: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_OremoveAll,type,
    removeAll: 
      !>[A: $tType] : ( A > fun(list(A),list(A)) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Orotate,type,
    rotate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles__rel,type,
    shuffles_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > $o ) ).

tff(sy_c_List_Osorted__wrt__rel,type,
    sorted_wrt_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).

tff(sy_c_List_Osplice,type,
    splice: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Osplice__rel,type,
    splice_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otake,type,
    take: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__add,type,
    map_add: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__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,bool) > 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(bool,A),product_prod(bool,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,bool),fun(list(A),list(A))) ).

tff(sy_c_Misc_Ofilter__rev__aux,type,
    filter_rev_aux: 
      !>[A: $tType] : ( list(A) > fun(fun(A,bool),fun(list(A),list(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__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))),bool)) ).

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)),bool)) ).

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,bool)) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge,type,
    merges9089515139780605204_merge: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * 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,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).

tff(sy_c_Misc_Omergesort__by__rel__rel,type,
    mergesort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).

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)),bool)) ).

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)),bool)) ).

tff(sy_c_Misc_Opartition__rev,type,
    partition_rev: 
      !>[A: $tType] : ( ( fun(A,bool) * 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,bool),product_prod(product_prod(list(A),list(A)),list(A))),fun(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool)) ).

tff(sy_c_Misc_Oquicksort__by__rel,type,
    quicksort_by_rel: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * 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,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),bool) ) > 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)),bool)) ).

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_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))),bool)) ).

tff(sy_c_Multiset_Oadd__mset,type,
    add_mset: 
      !>[A: $tType] : fun(A,fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
    comm_m7189776963980413722m_mset: 
      !>[A: $tType] : ( multiset(A) > A ) ).

tff(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
    comm_m9189036328036947845d_mset: 
      !>[A: $tType] : ( multiset(A) > A ) ).

tff(sy_c_Multiset_Ofilter__mset,type,
    filter_mset: 
      !>[A: $tType] : fun(fun(A,bool),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Ofold__mset,type,
    fold_mset: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * multiset(A) ) > B ) ).

tff(sy_c_Multiset_Oimage__mset,type,
    image_mset: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(multiset(A),multiset(B)) ) ).

tff(sy_c_Multiset_Ointer__mset,type,
    inter_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Olinorder__class_Opart,type,
    linorder_part: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).

tff(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset,type,
    linord6283353356039996273ltiset: 
      !>[A: $tType] : ( multiset(A) > list(A) ) ).

tff(sy_c_Multiset_Oms__strict,type,
    ms_strict: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).

tff(sy_c_Multiset_Oms__weak,type,
    ms_weak: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).

tff(sy_c_Multiset_Omset,type,
    mset: 
      !>[A: $tType] : ( 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,bool)) * 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,bool)) > fun(multiset(A),fun(multiset(A),bool)) ) ).

tff(sy_c_Multiset_Omultp__code,type,
    multp_code: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * multiset(A) * multiset(A) ) > $o ) ).

tff(sy_c_Multiset_Opcr__multiset,type,
    pcr_multiset: 
      !>[C: $tType,B: $tType] : ( fun(C,fun(B,bool)) > fun(fun(C,nat),fun(multiset(B),bool)) ) ).

tff(sy_c_Multiset_Opw__leq,type,
    pw_leq: ( multiset(product_prod(nat,nat)) * multiset(product_prod(nat,nat)) ) > $o ).

tff(sy_c_Multiset_Orel__mset,type,
    rel_mset: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(multiset(A),fun(multiset(B),bool)) ) ).

tff(sy_c_Multiset_Orepeat__mset,type,
    repeat_mset: 
      !>[A: $tType] : fun(nat,fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Oreplicate__mset,type,
    replicate_mset: 
      !>[A: $tType] : ( ( nat * A ) > multiset(A) ) ).

tff(sy_c_Multiset_Oset__mset,type,
    set_mset: 
      !>[A: $tType] : fun(multiset(A),set(A)) ).

tff(sy_c_Multiset_Osize__multiset,type,
    size_multiset: 
      !>[A: $tType] : ( fun(A,nat) > fun(multiset(A),nat) ) ).

tff(sy_c_Multiset_Osubset__eq__mset__impl,type,
    subset_eq_mset_impl: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > option(bool) ) ).

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)),bool)) ).

tff(sy_c_Multiset_Osubset__mset,type,
    subset_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),bool)) ).

tff(sy_c_Multiset_Osubseteq__mset,type,
    subseteq_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),bool)) ).

tff(sy_c_Multiset_Ounion__mset,type,
    union_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Owcount,type,
    wcount: 
      !>[A: $tType] : ( ( fun(A,nat) * multiset(A) ) > fun(A,nat) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,bool) ) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun(num,num) ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Option_Ocombine__options,type,
    combine_options: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * option(A) * option(A) ) > option(A) ) ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).

tff(sy_c_Option_Ooption_Orec__option,type,
    rec_option: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,C) ) > fun(option(A),C) ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( fun(A,nat) > fun(option(A),nat) ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),A) ).

tff(sy_c_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

tff(sy_c_Order__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_Olinear__order__on,type,
    order_679001287576687338der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Oofilter,type,
    order_ofilter: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).

tff(sy_c_Order__Relation_Opartial__order__on,type,
    order_7125193373082350890der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Opreorder__on,type,
    order_preorder_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Orelation__of,type,
    order_relation_of: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Order__Relation_Ounder,type,
    order_under: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(A,set(A)) ) ).

tff(sy_c_Order__Relation_OunderS,type,
    order_underS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Owell__order__on,type,
    order_well_order_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord_OLeast,type,
    least: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,bool),A) ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder_OGreatest,type,
    greatest: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,bool),A) ) ).

tff(sy_c_Orderings_Oorder_Omono,type,
    mono: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,B),bool) ) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),bool) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * 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 * set(A) ) > A ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : fun(A,fun(nat,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)),bool)) ).

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(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Omap__prod,type,
    product_map_prod: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
    product_rec_set_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
    product_rec_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > T ) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Product__Type_Oprod_Oswap,type,
    product_swap: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).

tff(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_Oscomp,type,
    product_scomp: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).

tff(sy_c_Product__Type_Ounit_OAbs__unit,type,
    product_Abs_unit: fun(bool,product_unit) ).

tff(sy_c_Product__Type_Ounit_ORep__unit,type,
    product_Rep_unit: fun(product_unit,bool) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
    linorder_quicksort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
    linord6200660962353139674rt_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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)),bool)) ).

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),bool)) ).

tff(sy_c_Random_Ominus__shift,type,
    minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Onext,type,
    next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Opick,type,
    pick: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(code_natural,A) ) ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Oselect,type,
    select: 
      !>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Oselect__weight,type,
    select_weight: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Osplit__seed,type,
    split_seed: product_prod(code_natural,code_natural) > product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun(product_prod(int,int),rat) ).

tff(sy_c_Rat_OFract,type,
    fract: fun(int,fun(int,rat)) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ocr__rat,type,
    cr_rat: ( product_prod(int,int) * rat ) > $o ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : fun(rat,A) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,bool)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,bool) ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_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_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_Relation_ODomainp,type,
    domainp: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(A,bool)) ).

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)) > fun(set(A),set(B)) ) ).

tff(sy_c_Relation_ORangep,type,
    rangep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,bool) ) ).

tff(sy_c_Relation_Oantisym,type,
    antisym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oasymp,type,
    asymp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Oconverse,type,
    converse: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).

tff(sy_c_Relation_Oconversep,type,
    conversep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,fun(A,bool)) ) ).

tff(sy_c_Relation_Oinv__image,type,
    inv_image: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Orefl__on,type,
    refl_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

tff(sy_c_Relation_Orelcompp,type,
    relcompp: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool)))) ).

tff(sy_c_Relation_Osingle__valued,type,
    single_valued: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

tff(sy_c_Relation_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_Rings_Oalgebraic__semidom__class_Ocoprime,type,
    algebr8660921524188924756oprime: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun(bool,A) ).

tff(sy_c_Set_OBall,type,
    ball: 
      !>[A: $tType] : fun(set(A),fun(fun(A,bool),bool)) ).

tff(sy_c_Set_OBex,type,
    bex: 
      !>[A: $tType] : fun(set(A),fun(fun(A,bool),bool)) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,bool),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : fun(set(A),fun(set(A),bool)) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,bool) * 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,
    insert: 
      !>[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,bool)) * set(A) ) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set_Ovimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),fun(set(B),set(A))) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool)) ).

tff(sy_c_Set__Interval_Oord_OatLeast,type,
    set_atLeast: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
    set_atLeastAtMost: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
    set_atLeastLessThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OatMost,type,
    set_atMost: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OgreaterThan,type,
    set_greaterThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
    set_gr3752724095348155675AtMost: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
    set_gr287244882034783167ssThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord_OlessThan,type,
    set_lessThan: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_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,bool)) > $o ) ).

tff(sy_c_Transfer_Obi__unique,type,
    bi_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oleft__total,type,
    left_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oleft__unique,type,
    left_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oright__total,type,
    right_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oright__unique,type,
    right_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Otransfer__bforall,type,
    transfer_bforall: 
      !>[A: $tType] : ( ( fun(A,bool) * fun(A,bool) ) > $o ) ).

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Typedef_Otype__definition,type,
    type_definition: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,bool) ) ).

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,bool)) > fun(set(A),fun(set(A),bool)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Wellfounded_OwfP,type,
    wfP: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Zorn_OChains,type,
    chains: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).

tff(sy_c_Zorn_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,bool)) ) > fun(set(A),bool) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( A > fun(set(A),bool) ) ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_P,type,
    p: assn ).

tff(sy_v_Q,type,
    q: fun(a,assn) ).

tff(sy_v_R,type,
    r: assn ).

tff(sy_v_as1____,type,
    as1: set(nat) ).

tff(sy_v_as2____,type,
    as2: set(nat) ).

tff(sy_v_as____,type,
    as: set(nat) ).

tff(sy_v_c,type,
    c: heap_Time_Heap(a) ).

tff(sy_v_h_H____,type,
    h: heap_ext(product_unit) ).

tff(sy_v_h____,type,
    h2: heap_ext(product_unit) ).

tff(sy_v_r____,type,
    r2: a ).

tff(sy_v_t____,type,
    t: nat ).

% Relevant facts (9324)
tff(fact_0_DJ,axiom,
    aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),as1),as2) = bot_bot(set(nat)) ).

% DJ
tff(fact_1__092_060open_062_Ih_H_M_Aas2_J_A_092_060Turnstile_062_AR_092_060close_062,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),h),as2))) ).

% \<open>(h', as2) \<Turnstile> R\<close>
tff(fact_2_M2,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),h2),as2))) ).

% M2
tff(fact_3_DJN,axiom,
    aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),hoare_new_addrs(h2,as1,h)),as2) = bot_bot(set(nat)) ).

% DJN
tff(fact_4__092_060open_062new__addrs_Ah_Aas_Ah_H_A_061_Anew__addrs_Ah_Aas1_Ah_H_A_092_060union_062_Aas2_092_060close_062,axiom,
    hoare_new_addrs(h2,as,h) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),hoare_new_addrs(h2,as1,h)),as2) ).

% \<open>new_addrs h as h' = new_addrs h as1 h' \<union> as2\<close>
tff(fact_5_new__addr__refl,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] : hoare_new_addrs(H,As,H) = As ).

% new_addr_refl
tff(fact_6_M1,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(p),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),h2),as1))) ).

% M1
tff(fact_7_MDL,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(a,assn,q,r2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),hoare_new_addrs(h2,as1,h)))) ).

% MDL
tff(fact_8__092_060open_062_Ih_M_Aas_J_A_092_060Turnstile_062_AP_A_K_AR_092_060close_062,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),p),r)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),h2),as))) ).

% \<open>(h, as) \<Turnstile> P * R\<close>
tff(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062as1_Aas2_O_A_092_060lbrakk_062as_A_061_Aas1_A_092_060union_062_Aas2_059_Aas1_A_092_060inter_062_Aas2_A_061_A_123_125_059_A_Ih_M_Aas1_J_A_092_060Turnstile_062_AP_059_A_Ih_M_Aas2_J_A_092_060Turnstile_062_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [As1: set(nat),As2: set(nat)] :
        ( ( as = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As2) )
       => ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As2) = bot_bot(set(nat)) )
         => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(p),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),h2),As1)))
           => ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),h2),As2))) ) ) ) ).

% \<open>\<And>thesis. (\<And>as1 as2. \<lbrakk>as = as1 \<union> as2; as1 \<inter> as2 = {}; (h, as1) \<Turnstile> P; (h, as2) \<Turnstile> R\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_10_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( rep_assn(X) = rep_assn(Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_11_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_iff(5)
tff(fact_12__092_060open_062relH_Aas2_Ah_Ah_H_092_060close_062,axiom,
    relH(as2,h2,h) ).

% \<open>relH as2 h h'\<close>
tff(fact_13_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_14_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
    <=> ( ( A3 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_15_mod__starD,axiom,
    ! [A5: assn,B4: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B4)),H))
     => ? [H1: product_prod(heap_ext(product_unit),set(nat)),H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(A5),H1))
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(B4),H2)) ) ) ).

% mod_starD
tff(fact_16_mod__starE,axiom,
    ! [A3: assn,B2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),H))
     => ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(A3),X_1))
         => ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(B2),H_2)) ) ) ).

% mod_starE
tff(fact_17_assms,axiom,
    hoare_hoare_triple(a,p,c,q) ).

% assms
tff(fact_18__092_060open_062lim_Ah_A_092_060le_062_Alim_Ah_H_092_060close_062,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,h2)),lim(product_unit,h))) ).

% \<open>lim h \<le> lim h'\<close>
tff(fact_19__092_060open_062as_A_061_Aas1_A_092_060union_062_Aas2_092_060close_062,axiom,
    as = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),as1),as2) ).

% \<open>as = as1 \<union> as2\<close>
tff(fact_20_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),bot_bot(set(nat))))) ) ).

% mod_h_bot_indep
tff(fact_21_relH__dist__union,axiom,
    ! [As: set(nat),As3: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As3),H,H3)
    <=> ( relH(As,H,H3)
        & relH(As3,H,H3) ) ) ).

% relH_dist_union
tff(fact_22_snga__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R: array(A),X3: list(A),H4: heap_ext(product_unit),As4: set(nat)] : X != 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)))),R),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))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))) ) ).

% snga_assn_raw.cases
tff(fact_23_sngr__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R: ref(A),X3: A,H4: heap_ext(product_unit),As4: set(nat)] : X != 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)))),R),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))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))) ) ).

% sngr_assn_raw.cases
tff(fact_24_times__assn__raw_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))] :
      ~ ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H4: heap_ext(product_unit),As4: set(nat)] : X != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),P2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))) ).

% times_assn_raw.cases
tff(fact_25_relH__sym,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(As,H,H3)
     => relH(As,H3,H) ) ).

% relH_sym
tff(fact_26_relH__trans,axiom,
    ! [As: set(nat),H12: heap_ext(product_unit),H22: heap_ext(product_unit),H32: heap_ext(product_unit)] :
      ( relH(As,H12,H22)
     => ( relH(As,H22,H32)
       => relH(As,H12,H32) ) ) ).

% relH_trans
tff(fact_27_pure__assn__raw_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(bool,product_prod(A,set(B)))] :
      ~ ! [B5: bool,H4: A,As4: set(B)] : X != aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),B5),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4)) ).

% pure_assn_raw.cases
tff(fact_28_mod__relH,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),P: assn] :
      ( relH(As,H,H3)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
      <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))) ) ) ).

% mod_relH
tff(fact_29_hoare__triple__preI,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H4))
         => hoare_hoare_triple(A,P,C2,Q) )
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% hoare_triple_preI
tff(fact_30_mod__star__conv,axiom,
    ! [A5: assn,B4: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B4)),H))
    <=> ? [Hr: heap_ext(product_unit),As12: set(nat),As22: set(nat)] :
          ( ( H = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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)),As12),As22)) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As22) = bot_bot(set(nat)) )
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(A5),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),As12)))
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(B4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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_31_star__assnI,axiom,
    ! [P: assn,H: heap_ext(product_unit),As: set(nat),Q: assn,As3: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As3)))
       => ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As3) = bot_bot(set(nat)) )
         => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As3)))) ) ) ) ).

% star_assnI
tff(fact_32_prod__induct7,axiom,
    ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B5),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C3),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2))))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),bool,P,X)) ) ).

% prod_induct7
tff(fact_33_prod__induct6,axiom,
    ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B5),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C3),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2)))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool,P,X)) ) ).

% prod_induct6
tff(fact_34_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A6: A,B5: B,C3: C,D2: D,E2: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A6),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B5),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P,X)) ) ).

% prod_induct5
tff(fact_35_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool),X: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A6: A,B5: B,C3: C,D2: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A6),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B5),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2)))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P,X)) ) ).

% prod_induct4
tff(fact_36_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),bool),X: product_prod(A,product_prod(B,C))] :
      ( ! [A6: A,B5: B,C3: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A6),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B5),C3))))
     => pp(aa(product_prod(A,product_prod(B,C)),bool,P,X)) ) ).

% prod_induct3
tff(fact_37_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B5),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C3),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))) ).

% prod_cases7
tff(fact_38_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B5),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C3),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))) ).

% prod_cases6
tff(fact_39_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A6: A,B5: B,C3: C,D2: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A6),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B5),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).

% prod_cases5
tff(fact_40_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A6: A,B5: B,C3: C,D2: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A6),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B5),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))) ).

% prod_cases4
tff(fact_41_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A6: A,B5: B,C3: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A6),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B5),C3)) ).

% prod_cases3
tff(fact_42_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
     => ~ ( ( A3 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_43_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool),P3: product_prod(A,B)] :
      ( ! [A6: A,B5: B] : pp(aa(product_prod(A,B),bool,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)))
     => pp(aa(product_prod(A,B),bool,P,P3)) ) ).

% prod_cases
tff(fact_44_surj__pair,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B)] :
    ? [X3: A,Y3: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) ).

% surj_pair
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(fun(A,bool),set(A),collect(A),P)))
    <=> pp(aa(A,bool,P,A3)) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [A: $tType,A5: set(A)] : aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)) = A5 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( aa(fun(A,bool),set(A),collect(A),P) = aa(fun(A,bool),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_48_ext,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F3,X3) = aa(A,B,G3,X3)
     => ( F3 = G3 ) ) ).

% ext
tff(fact_49_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A6: A,B5: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) ).

% old.prod.exhaust
tff(fact_50_one__assn__raw_Ocases,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ~ ! [H4: heap_ext(product_unit),As4: set(nat)] : X != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) ).

% one_assn_raw.cases
tff(fact_51_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),R2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),R2)) ).

% assn_times_assoc
tff(fact_52_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),P) ).

% assn_times_comm
tff(fact_53_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T2: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_54_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = S ).

% Int_Un_eq(3)
tff(fact_55_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_56_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_57_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T2: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_58_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)) = S ).

% Un_Int_eq(3)
tff(fact_59_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_60_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_61_Un__empty,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = bot_bot(set(A)) )
    <=> ( ( A5 = bot_bot(set(A)) )
        & ( B4 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_62_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_63_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_64_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).

% sup_bot_left
tff(fact_65_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),P) )
    <=> ! [X4: A] : ~ pp(aa(A,bool,P,X4)) ) ).

% empty_Collect_eq
tff(fact_66_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( aa(fun(A,bool),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ! [X4: A] : ~ pp(aa(A,bool,P,X4)) ) ).

% Collect_empty_eq
tff(fact_67_all__not__in__conv,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X4: A] : ~ pp(aa(set(A),bool,member(A,X4),A5))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_68_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ pp(aa(set(A),bool,member(A,C2),bot_bot(set(A)))) ).

% empty_iff
tff(fact_69_inf__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: 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)),F3),G3),X) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,X)),aa(A,B,G3,X)) ) ).

% inf_apply
tff(fact_70_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).

% inf_right_idem
tff(fact_71_inf_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).

% inf.right_idem
tff(fact_72_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).

% inf_left_idem
tff(fact_73_inf_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).

% inf.left_idem
tff(fact_74_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),X) = X ) ).

% inf_idem
tff(fact_75_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_76_sup__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: 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)),F3),G3),X) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,X)),aa(A,B,G3,X)) ) ).

% sup_apply
tff(fact_77_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.right_idem
tff(fact_78_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).

% sup_left_idem
tff(fact_79_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.left_idem
tff(fact_80_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),X) = X ) ).

% sup_idem
tff(fact_81_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_82_Int__iff,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
    <=> ( pp(aa(set(A),bool,member(A,C2),A5))
        & pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% Int_iff
tff(fact_83_IntI,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),A5))
     => ( pp(aa(set(A),bool,member(A,C2),B4))
       => pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ).

% IntI
tff(fact_84_Un__iff,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))
    <=> ( pp(aa(set(A),bool,member(A,C2),A5))
        | pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% Un_iff
tff(fact_85_UnCI,axiom,
    ! [A: $tType,C2: A,B4: set(A),A5: set(A)] :
      ( ( ~ pp(aa(set(A),bool,member(A,C2),B4))
       => pp(aa(set(A),bool,member(A,C2),A5)) )
     => pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))) ) ).

% UnCI
tff(fact_86_mod__or__dist,axiom,
    ! [P: assn,Q: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q)),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H))
        | pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),H)) ) ) ).

% mod_or_dist
tff(fact_87_star__false__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P) = bot_bot(assn) ).

% star_false_left
tff(fact_88_star__false__right,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),bot_bot(assn)) = bot_bot(assn) ).

% star_false_right
tff(fact_89_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% inf.bounded_iff
tff(fact_90_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).

% le_inf_iff
tff(fact_91_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% sup.bounded_iff
tff(fact_92_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% le_sup_iff
tff(fact_93_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_94_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_95_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_96_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_97_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_98_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = bot_bot(A) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_99_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = bot_bot(A) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_100_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_101_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% sup_bot_right
tff(fact_102_mod__h__bot__iff_I6_J,axiom,
    ! [P: assn,Q: assn,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_iff(6)
tff(fact_103_mod__h__bot__iff_I7_J,axiom,
    ! [P: assn,Q: assn,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
        | pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_iff(7)
tff(fact_104_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) ).

% bot_set_def
tff(fact_105_mod__and__dist,axiom,
    ! [P: assn,Q: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),H)) ) ) ).

% mod_and_dist
tff(fact_106_mod__false,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(bot_bot(assn)),H)) ).

% mod_false
tff(fact_107_star__or__dist1,axiom,
    ! [A5: assn,B4: assn,C4: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B4)),C4) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),C4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B4),C4)) ).

% star_or_dist1
tff(fact_108_star__or__dist2,axiom,
    ! [C4: assn,A5: assn,B4: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B4)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),A5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),B4)) ).

% star_or_dist2
tff(fact_109_ex__in__conv,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ? [X4: A] : pp(aa(set(A),bool,member(A,X4),A5))
    <=> ( A5 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_110_equals0I,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [Y3: A] : ~ pp(aa(set(A),bool,member(A,Y3),A5))
     => ( A5 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_111_equals0D,axiom,
    ! [A: $tType,A5: set(A),A3: A] :
      ( ( A5 = bot_bot(set(A)) )
     => ~ pp(aa(set(A),bool,member(A,A3),A5)) ) ).

% equals0D
tff(fact_112_emptyE,axiom,
    ! [A: $tType,A3: A] : ~ pp(aa(set(A),bool,member(A,A3),bot_bot(set(A)))) ).

% emptyE
tff(fact_113_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X5: 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)),F3),G3),X5) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,X5)),aa(A,B,G3,X5)) ) ).

% inf_fun_def
tff(fact_114_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).

% inf_left_commute
tff(fact_115_inf_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.left_commute
tff(fact_116_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X) ) ).

% inf_commute
tff(fact_117_inf_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),A3) ) ).

% inf.commute
tff(fact_118_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).

% inf_assoc
tff(fact_119_inf_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.assoc
tff(fact_120_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X) ) ).

% inf_sup_aci(1)
tff(fact_121_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).

% inf_sup_aci(2)
tff(fact_122_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).

% inf_sup_aci(3)
tff(fact_123_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).

% inf_sup_aci(4)
tff(fact_124_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X5: 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)),F3),G3),X5) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,X5)),aa(A,B,G3,X5)) ) ).

% sup_fun_def
tff(fact_125_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).

% sup_left_commute
tff(fact_126_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.left_commute
tff(fact_127_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).

% sup_commute
tff(fact_128_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),A3) ) ).

% sup.commute
tff(fact_129_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).

% sup_assoc
tff(fact_130_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.assoc
tff(fact_131_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).

% inf_sup_aci(5)
tff(fact_132_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).

% inf_sup_aci(6)
tff(fact_133_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).

% inf_sup_aci(7)
tff(fact_134_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).

% inf_sup_aci(8)
tff(fact_135_Int__left__commute,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4)) ).

% Int_left_commute
tff(fact_136_Int__left__absorb,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ).

% Int_left_absorb
tff(fact_137_Int__commute,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),A5) ).

% Int_commute
tff(fact_138_Int__absorb,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),A5) = A5 ).

% Int_absorb
tff(fact_139_Int__assoc,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) ).

% Int_assoc
tff(fact_140_IntD2,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
     => pp(aa(set(A),bool,member(A,C2),B4)) ) ).

% IntD2
tff(fact_141_IntD1,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
     => pp(aa(set(A),bool,member(A,C2),A5)) ) ).

% IntD1
tff(fact_142_IntE,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
     => ~ ( pp(aa(set(A),bool,member(A,C2),A5))
         => ~ pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% IntE
tff(fact_143_Un__left__commute,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C4)) ).

% Un_left_commute
tff(fact_144_Un__left__absorb,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) ).

% Un_left_absorb
tff(fact_145_Un__commute,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),A5) ).

% Un_commute
tff(fact_146_Un__absorb,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),A5) = A5 ).

% Un_absorb
tff(fact_147_Un__assoc,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4)) ).

% Un_assoc
tff(fact_148_ball__Un,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))
         => pp(aa(A,bool,P,X4)) )
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(A,bool,P,X4)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B4))
           => pp(aa(A,bool,P,X4)) ) ) ) ).

% ball_Un
tff(fact_149_bex__Un,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))
          & pp(aa(A,bool,P,X4)) )
    <=> ( ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
            & pp(aa(A,bool,P,X4)) )
        | ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B4))
            & pp(aa(A,bool,P,X4)) ) ) ) ).

% bex_Un
tff(fact_150_UnI2,axiom,
    ! [A: $tType,C2: A,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),B4))
     => pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))) ) ).

% UnI2
tff(fact_151_UnI1,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),A5))
     => pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))) ) ).

% UnI1
tff(fact_152_UnE,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))
     => ( ~ pp(aa(set(A),bool,member(A,C2),A5))
       => pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% UnE
tff(fact_153_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.coboundedI2
tff(fact_154_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.coboundedI1
tff(fact_155_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_156_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb_iff1
tff(fact_157_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2)) ) ).

% inf.cobounded2
tff(fact_158_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3)) ) ).

% inf.cobounded1
tff(fact_159_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.order_iff
tff(fact_160_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))) ) ) ) ).

% inf_greatest
tff(fact_161_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))) ) ) ) ).

% inf.boundedI
tff(fact_162_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% inf.boundedE
tff(fact_163_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_164_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_165_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_166_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb1
tff(fact_167_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_168_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),X3))
         => ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Y3))
           => ( ! [X3: A,Y3: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_169_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% inf.orderI
tff(fact_170_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.orderE
tff(fact_171_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI2
tff(fact_172_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI1
tff(fact_173_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D3))) ) ) ) ).

% inf_mono
tff(fact_174_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))) ) ) ) ).

% le_infI
tff(fact_175_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).

% le_infE
tff(fact_176_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_le2
tff(fact_177_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_le1
tff(fact_178_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_sup_ord(1)
tff(fact_179_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_sup_ord(2)
tff(fact_180_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI2
tff(fact_181_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI1
tff(fact_182_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_183_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb_iff1
tff(fact_184_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded2
tff(fact_185_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded1
tff(fact_186_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.order_iff
tff(fact_187_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)) ) ) ) ).

% sup.boundedI
tff(fact_188_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% sup.boundedE
tff(fact_189_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_190_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_191_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_192_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb1
tff(fact_193_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)))
         => ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)))
           => ( ! [X3: A,Y3: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3)),X3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_194_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% sup.orderI
tff(fact_195_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.orderE
tff(fact_196_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_197_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X)) ) ) ) ).

% sup_least
tff(fact_198_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3))) ) ) ) ).

% sup_mono
tff(fact_199_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,D3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ) ).

% sup.mono
tff(fact_200_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI2
tff(fact_201_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI1
tff(fact_202_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge2
tff(fact_203_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge1
tff(fact_204_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)) ) ) ) ).

% le_supI
tff(fact_205_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).

% le_supE
tff(fact_206_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(3)
tff(fact_207_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(4)
tff(fact_208_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),B4))
             => ( X4 != Xa ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_209_Int__empty__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_210_Int__empty__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B4) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_211_disjoint__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ~ pp(aa(set(A),bool,member(A,X4),B4)) ) ) ).

% disjoint_iff
tff(fact_212_Int__emptyI,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => ~ pp(aa(set(A),bool,member(A,X3),B4)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_213_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).

% sup_inf_distrib2
tff(fact_214_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).

% sup_inf_distrib1
tff(fact_215_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).

% inf_sup_distrib2
tff(fact_216_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).

% inf_sup_distrib1
tff(fact_217_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Z3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ) ) ).

% distrib_imp2
tff(fact_218_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Z3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ) ) ).

% distrib_imp1
tff(fact_219_Un__empty__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),bot_bot(set(A))) = A5 ).

% Un_empty_right
tff(fact_220_Un__empty__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B4) = B4 ).

% Un_empty_left
tff(fact_221_Un__Int__distrib2,axiom,
    ! [A: $tType,B4: set(A),C4: set(A),A5: 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)),B4),C4)),A5) = 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)),B4),A5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A5)) ).

% Un_Int_distrib2
tff(fact_222_Int__Un__distrib2,axiom,
    ! [A: $tType,B4: set(A),C4: set(A),A5: 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)),B4),C4)),A5) = 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)),B4),A5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A5)) ).

% Int_Un_distrib2
tff(fact_223_Un__Int__distrib,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) = 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C4)) ).

% Un_Int_distrib
tff(fact_224_Int__Un__distrib,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4)) = 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4)) ).

% Int_Un_distrib
tff(fact_225_Un__Int__crazy,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A5)) = 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A5)) ).

% Un_Int_crazy
tff(fact_226_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)))) ) ).

% distrib_sup_le
tff(fact_227_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)))) ) ).

% distrib_inf_le
tff(fact_228_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_229_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_230_mod__h__bot__iff_I4_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Q3: array(B),Y: list(B),H: heap_ext(product_unit)] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(snga_assn(B,Q3,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)),H),bot_bot(set(nat))))) ) ).

% mod_h_bot_iff(4)
tff(fact_231_mod__h__bot__iff_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),X: A,H: heap_ext(product_unit)] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(sngr_assn(A,P3,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),bot_bot(set(nat))))) ) ).

% mod_h_bot_iff(3)
tff(fact_232_old_Oprod_Orec,axiom,
    ! [A: $tType,T: $tType,B: $tType,F1: fun(A,fun(B,T)),A3: A,B2: B] : product_rec_prod(A,B,T,F1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) = aa(B,T,aa(A,fun(B,T),F1,A3),B2) ).

% old.prod.rec
tff(fact_233_EX,axiom,
    aa(heap_ext(product_unit),option(product_prod(a,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(a,c),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),h),t))) ).

% EX
tff(fact_234_bot__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( bot(C)
     => ! [X: D] : aa(D,C,bot_bot(fun(D,C)),X) = bot_bot(C) ) ).

% bot_apply
tff(fact_235_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ? [As13: set(nat),As23: set(nat)] :
                ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As23) )
                & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As23) = bot_bot(set(nat)) )
                & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As13)))
                & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),As23))) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_236_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ~ ? [As1: set(nat),As2: set(nat)] :
                  ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As2) )
                  & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As2) = bot_bot(set(nat)) )
                  & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As1)))
                  & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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))) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_237_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb))
      <=> pp(Y) )
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ( pp(Y)
            <=> ~ ? [As12: set(nat),As22: set(nat)] :
                    ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As22) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As22) = bot_bot(set(nat)) )
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As12)))
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),As22))) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_238_times__assn__raw_Osimps,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
    <=> ? [As12: set(nat),As22: set(nat)] :
          ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As22) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As22) = bot_bot(set(nat)) )
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As12)))
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22))) ) ) ).

% times_assn_raw.simps
tff(fact_239_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).

% dual_order.refl
tff(fact_240_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).

% order_refl
tff(fact_241_subset__empty,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),bot_bot(set(A))))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_242_empty__subsetI,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),bot_bot(set(A))),A5)) ).

% empty_subsetI
tff(fact_243_Int__subset__iff,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A5))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B4)) ) ) ).

% Int_subset_iff
tff(fact_244_Un__subset__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)),C4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4)) ) ) ).

% Un_subset_iff
tff(fact_245_sngr__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),X: A,Y: A] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P3,X)),sngr_assn(A,P3,Y)) = bot_bot(assn) ) ).

% sngr_same_false
tff(fact_246_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),X: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,P3,X)),snga_assn(A,P3,Y)) = bot_bot(assn) ) ).

% snga_same_false
tff(fact_247_less__eq__assn__def,axiom,
    ! [A3: assn,B2: assn] :
      ( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less_eq(assn),A3),B2))
    <=> ( A3 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),B2) ) ) ).

% less_eq_assn_def
tff(fact_248_Int__mono,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B4: set(A),D4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),D4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),D4))) ) ) ).

% Int_mono
tff(fact_249_Int__lower1,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)),A5)) ).

% Int_lower1
tff(fact_250_Int__lower2,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)),B4)) ).

% Int_lower2
tff(fact_251_Int__absorb1,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_252_Int__absorb2,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = A5 ) ) ).

% Int_absorb2
tff(fact_253_Int__greatest,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ).

% Int_greatest
tff(fact_254_Int__Collect__mono,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => ( pp(aa(A,bool,P,X3))
             => pp(aa(A,bool,Q,X3)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,bool),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(fun(A,bool),set(A),collect(A),Q)))) ) ) ).

% Int_Collect_mono
tff(fact_255_Un__mono,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B4: set(A),D4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),D4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),D4))) ) ) ).

% Un_mono
tff(fact_256_Un__least,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)),C4)) ) ) ).

% Un_least
tff(fact_257_Un__upper1,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))) ).

% Un_upper1
tff(fact_258_Un__upper2,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))) ).

% Un_upper2
tff(fact_259_Un__absorb1,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_260_Un__absorb2,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = A5 ) ) ).

% Un_absorb2
tff(fact_261_subset__UnE,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))
     => ~ ! [A7: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A7),A5))
           => ! [B6: set(A)] :
                ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),B4))
               => ( C4 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A7),B6) ) ) ) ) ).

% subset_UnE
tff(fact_262_subset__Un__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_263_relH__subset,axiom,
    ! [Bs: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),As: set(nat)] :
      ( relH(Bs,H,H3)
     => ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As),Bs))
       => relH(As,H,H3) ) ) ).

% relH_subset
tff(fact_264_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A5)) ) ).

% Un_Int_assoc_eq
tff(fact_265_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ( B2 != A3 ) ) ) ) ).

% nle_le
tff(fact_266_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) )
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
               => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
                   => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X)) )
                 => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_267_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% order_class.order_eq_iff
tff(fact_268_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% ord_eq_le_trans
tff(fact_269_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% ord_le_eq_trans
tff(fact_270_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_271_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% order.trans
tff(fact_272_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).

% order_trans
tff(fact_273_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A3: A,B2: A] :
          ( ! [A6: A,B5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B5)) )
         => ( ! [A6: A,B5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),P,B5),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B5)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),P,A3),B2)) ) ) ) ).

% linorder_wlog
tff(fact_274_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% dual_order.eq_iff
tff(fact_275_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( A3 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_276_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% dual_order.trans
tff(fact_277_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => ( A3 = B2 ) ) ) ) ).

% antisym
tff(fact_278_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))) ) ) ).

% le_funD
tff(fact_279_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))) ) ) ).

% le_funE
tff(fact_280_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)))
         => pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3)) ) ) ).

% le_funI
tff(fact_281_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
        <=> ! [X4: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4))) ) ) ).

% le_fun_def
tff(fact_282_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_283_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,C2))) ) ) ) ) ).

% order_subst1
tff(fact_284_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,A3)),C2)) ) ) ) ) ).

% order_subst2
tff(fact_285_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_eq_refl
tff(fact_286_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_linear
tff(fact_287_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C2: B] :
          ( ( A3 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,C2))) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_288_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F3: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( aa(A,B,F3,B2) = C2 )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),C2)) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_289_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_le_cases
tff(fact_290_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_291_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X5: A] : aa(A,B,bot_bot(fun(A,B)),X5) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_292_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: A,K: A,A3: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_293_boolean__algebra__cancel_Oinf2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B4: A,K: A,B2: A,A3: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B4) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_294_boolean__algebra__cancel_Osup1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: A,K: A,A3: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_295_boolean__algebra__cancel_Osup2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B4: A,K: A,B2: A,A3: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B4) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_296_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A3)) ) ).

% bot.extremum
tff(fact_297_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_298_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_299_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_300_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_301_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_302_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_303_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_304_sup__Some,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_Some
tff(fact_305_inf__Some,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) ) ).

% inf_Some
tff(fact_306_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% less_eq_option_Some
tff(fact_307__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_Ah_H_At_O_A_092_060lbrakk_062execute_Ac_Ah_A_061_ASome_A_Ir_M_Ah_H_M_At_J_059_A_Ih_H_M_Anew__addrs_Ah_Aas1_Ah_H_J_A_092_060Turnstile_062_AQ_Ar_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_Ah_Ah_H_059_Alim_Ah_A_092_060le_062_Alim_Ah_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [R: a,H5: 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,c),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)),R),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),H5),T3)))
       => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(a,assn,q,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),hoare_new_addrs(h2,as1,H5))))
         => ( relH(aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aa(nat,bool)),h2,H5)
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,h2)),lim(product_unit,H5))) ) ) ) ).

% \<open>\<And>thesis. (\<And>r h' t. \<lbrakk>execute c h = Some (r, h', t); (h', new_addrs h as1 h') \<Turnstile> Q r; relH {a. a < lim h \<and> a \<notin> as1} h h'; lim h \<le> lim h'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_308_option_Oinject,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X2) = aa(A,option(A),some(A),Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_309_bot__empty__eq,axiom,
    ! [A: $tType,X5: A] :
      ( pp(aa(A,bool,bot_bot(fun(A,bool)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),bot_bot(set(A)))) ) ).

% bot_empty_eq
tff(fact_310_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( aa(fun(A,bool),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,bool)) ) ) ).

% Collect_empty_eq_bot
tff(fact_311_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,X: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F2: fun(A,B),A6: A,B5: A] : X != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)) ).

% pairself.cases
tff(fact_312_disjoint__mono,axiom,
    ! [A: $tType,A3: set(A),A4: set(A),B2: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B2),B3))
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ) ) ).

% disjoint_mono
tff(fact_313_disjointI,axiom,
    ! [A: $tType,A3: set(A),B2: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A3))
         => ~ pp(aa(set(A),bool,member(A,X3),B2)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ).

% disjointI
tff(fact_314_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,X: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),M)),X))
         => ~ ! [M2: A] :
                ( ( X = aa(A,option(A),some(A),M2) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),M2)) ) ) ) ).

% le_some_optE
tff(fact_315_subsetI,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(set(A),bool,member(A,X3),B4)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ).

% subsetI
tff(fact_316_subset__antisym,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => ( A5 = B4 ) ) ) ).

% subset_antisym
tff(fact_317_RH1,axiom,
    relH(aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aa(nat,bool)),h2,h) ).

% RH1
tff(fact_318_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% less_option_Some
tff(fact_319__092_060open_062relH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_Ah_Ah_H_092_060close_062,axiom,
    relH(aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ab(nat,bool)),h2,h) ).

% \<open>relH {a. a < lim h \<and> a \<notin> as} h h'\<close>
tff(fact_320__092_060open_062as2_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
    pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),as2),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aa(nat,bool)))) ).

% \<open>as2 \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
tff(fact_321__092_060open_062_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
    pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ab(nat,bool))),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aa(nat,bool)))) ).

% \<open>{a. a < lim h \<and> a \<notin> as} \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
tff(fact_322_in__mono,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => pp(aa(set(A),bool,member(A,X),B4)) ) ) ).

% in_mono
tff(fact_323_subsetD,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,member(A,C2),A5))
       => pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% subsetD
tff(fact_324_equalityE,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( A5 = B4 )
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5)) ) ) ).

% equalityE
tff(fact_325_subset__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(set(A),bool,member(A,X4),B4)) ) ) ).

% subset_eq
tff(fact_326_equalityD1,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( A5 = B4 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ).

% equalityD1
tff(fact_327_equalityD2,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( A5 = B4 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5)) ) ).

% equalityD2
tff(fact_328_subset__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
    <=> ! [T4: A] :
          ( pp(aa(set(A),bool,member(A,T4),A5))
         => pp(aa(set(A),bool,member(A,T4),B4)) ) ) ).

% subset_iff
tff(fact_329_subset__refl,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),A5)) ).

% subset_refl
tff(fact_330_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q))) ) ).

% Collect_mono
tff(fact_331_subset__trans,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4)) ) ) ).

% subset_trans
tff(fact_332_set__eq__subset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( A5 = B4 )
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5)) ) ) ).

% set_eq_subset
tff(fact_333_sup__Un__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),R2)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R2),S))) ) ).

% sup_Un_eq
tff(fact_334_Collect__subset,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),A5),P))),A5)) ).

% Collect_subset
tff(fact_335_inf__Int__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),R2)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R2),S))) ) ).

% inf_Int_eq
tff(fact_336_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),S)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R2),S))) ) ).

% sup_Un_eq2
tff(fact_337_less__eq__set__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B4))) ) ).

% less_eq_set_def
tff(fact_338_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),S)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R2),S))) ) ).

% inf_Int_eq2
tff(fact_339_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q)))
    <=> ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
         => pp(aa(A,bool,Q,X4)) ) ) ).

% Collect_mono_iff
tff(fact_340_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),bot_bot(set(product_prod(A,B))))) ) ).

% bot_empty_eq2
tff(fact_341_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Xa: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa)),R2))
        <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa)),S)) )
    <=> ( R2 = S ) ) ).

% pred_equals_eq2
tff(fact_342_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),S)))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R2),S)) ) ).

% pred_subset_eq2
tff(fact_343_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X)) ) ).

% lt_ex
tff(fact_344_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_1)) ) ).

% gt_ex
tff(fact_345_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),Y)) ) ) ) ).

% dense
tff(fact_346_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_347_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order.asym
tff(fact_348_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% ord_eq_less_trans
tff(fact_349_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% ord_less_eq_trans
tff(fact_350_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3))
                 => pp(aa(A,bool,P,Y4)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% less_induct
tff(fact_351_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_352_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_cases
tff(fact_353_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% dual_order.asym
tff(fact_354_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).

% dual_order.irrefl
tff(fact_355_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
        <=> ? [N: A] :
              ( pp(aa(A,bool,P,N))
              & ! [M3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M3),N))
                 => ~ pp(aa(A,bool,P,M3)) ) ) ) ) ).

% exists_least_iff
tff(fact_356_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A3: A,B2: A] :
          ( ! [A6: A,B5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B5)) )
         => ( ! [A6: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,A6),A6))
           => ( ! [A6: A,B5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),P,B5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A3),B2)) ) ) ) ) ).

% linorder_less_wlog
tff(fact_357_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% order.strict_trans
tff(fact_358_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_359_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% dual_order.strict_trans
tff(fact_360_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( A3 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_361_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( A3 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_362_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE
tff(fact_363_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_asym
tff(fact_364_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neq_iff
tff(fact_365_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order_less_asym'
tff(fact_366_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).

% order_less_trans
tff(fact_367_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C2: B] :
          ( ( A3 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C2))) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_368_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F3: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( aa(A,B,F3,B2) = C2 )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,A3)),C2)) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_369_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).

% order_less_irrefl
tff(fact_370_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C2))) ) ) ) ) ).

% order_less_subst1
tff(fact_371_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C2)) ) ) ) ) ).

% order_less_subst2
tff(fact_372_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_not_sym
tff(fact_373_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => pp(P) ) ) ) ).

% order_less_imp_triv
tff(fact_374_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          | ( X = Y )
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_less_linear
tff(fact_375_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_376_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_377_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_imp_not_less
tff(fact_378_subset__Collect__conv,axiom,
    ! [A: $tType,S: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(fun(A,bool),set(A),collect(A),P)))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S))
         => pp(aa(A,bool,P,X4)) ) ) ).

% subset_Collect_conv
tff(fact_379_pred__subset__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),R2)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R2),S)) ) ).

% pred_subset_eq
tff(fact_380_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ae(A,bool)) ).

% Set.empty_def
tff(fact_381_Int__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_af(set(A),fun(set(A),fun(A,bool)),A5),B4)) ).

% Int_def
tff(fact_382_Int__Collect,axiom,
    ! [A: $tType,X: A,A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,bool),set(A),collect(A),P))))
    <=> ( pp(aa(set(A),bool,member(A,X),A5))
        & pp(aa(A,bool,P,X)) ) ) ).

% Int_Collect
tff(fact_383_inf__set__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B4))) ).

% inf_set_def
tff(fact_384_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ag(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q)) ).

% Collect_conj_eq
tff(fact_385_Un__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ah(set(A),fun(set(A),fun(A,bool)),A5),B4)) ).

% Un_def
tff(fact_386_sup__set__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B4))) ).

% sup_set_def
tff(fact_387_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ai(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q)) ).

% Collect_disj_eq
tff(fact_388_exists__leI,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ( ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),N2))
           => ~ pp(aa(nat,bool,P,N3)) )
       => pp(aa(nat,bool,P,N2)) )
     => ? [N4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N2))
          & pp(aa(nat,bool,P,N4)) ) ) ).

% exists_leI
tff(fact_389_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_390_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_le_less_linear
tff(fact_391_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C2)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_392_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C2))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_393_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C2)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_394_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C2))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_395_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).

% order_less_le_trans
tff(fact_396_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).

% order_le_less_trans
tff(fact_397_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_398_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( A3 != B2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_399_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_less_imp_le
tff(fact_400_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_not_less
tff(fact_401_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_not_le
tff(fact_402_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_403_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_404_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% dual_order.strict_implies_order
tff(fact_405_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% order.strict_implies_order
tff(fact_406_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_407_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_408_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_409_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_410_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
            | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_411_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z2)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% dense_le_bounded
tff(fact_412_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),X))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),W)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% dense_ge_bounded
tff(fact_413_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% order.strict_iff_not
tff(fact_414_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% order.strict_trans2
tff(fact_415_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% order.strict_trans1
tff(fact_416_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_417_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_418_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% not_le_imp_less
tff(fact_419_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% less_le_not_le
tff(fact_420_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z2: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).

% dense_le
tff(fact_421_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,Y: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).

% dense_ge
tff(fact_422_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_423_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_424_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% nless_le
tff(fact_425_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% leI
tff(fact_426_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% leD
tff(fact_427_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A3)) ) ) ).

% bot.not_eq_extremum
tff(fact_428_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),bot_bot(A))) ) ).

% bot.extremum_strict
tff(fact_429_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.strict_coboundedI2
tff(fact_430_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).

% inf.strict_coboundedI1
tff(fact_431_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_432_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% inf.strict_boundedE
tff(fact_433_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_434_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb3
tff(fact_435_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% less_infI2
tff(fact_436_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% less_infI1
tff(fact_437_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.strict_coboundedI2
tff(fact_438_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.strict_coboundedI1
tff(fact_439_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_440_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% sup.strict_boundedE
tff(fact_441_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_442_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb3
tff(fact_443_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% less_supI2
tff(fact_444_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% less_supI1
tff(fact_445_new__addrs__def,axiom,
    ! [H: heap_ext(product_unit),As: set(nat),H3: heap_ext(product_unit)] : hoare_new_addrs(H,As,H3) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),aa(fun(nat,bool),set(nat),collect(nat),aa(heap_ext(product_unit),fun(nat,bool),aTP_Lamp_aj(heap_ext(product_unit),fun(heap_ext(product_unit),fun(nat,bool)),H),H3))) ).

% new_addrs_def
tff(fact_446_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => ( ( C2 = D3 )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),D3)) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_447_bex2I,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,S: set(product_prod(A,B)),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
     => ( ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
         => pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) )
       => ? [A6: A,B5: B] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),S))
            & pp(aa(B,bool,aa(A,fun(B,bool),P,A6),B5)) ) ) ) ).

% bex2I
tff(fact_448_subrelI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X3: A,Y3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R3))
         => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),S2)) )
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S2)) ) ).

% subrelI
tff(fact_449_memb__imp__not__empty,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),S))
     => ( S != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_450_set__notEmptyE,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ~ ! [X3: A] : ~ pp(aa(set(A),bool,member(A,X3),S)) ) ).

% set_notEmptyE
tff(fact_451_inter__eq__subsetI,axiom,
    ! [A: $tType,S: set(A),S3: set(A),A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),S3))
     => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),S3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),S3) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_452_hoare__triple__def,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,C2,Q)
    <=> ! [H6: heap_ext(product_unit),As5: set(nat)] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As5)))
         => ? [R4: A,H7: heap_ext(product_unit)] :
              ( ? [T4: nat] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H6) = 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)),R4),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),H7),T4)))
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q,R4)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),hoare_new_addrs(H6,As5,H7))))
              & relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ak(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H6),As5)),H6,H7)
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H6)),lim(product_unit,H7))) ) ) ) ).

% hoare_triple_def
tff(fact_453_hoare__tripleI,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ! [H4: heap_ext(product_unit),As4: set(nat)] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
         => ? [R5: A,H8: heap_ext(product_unit),T5: nat] :
              ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H4) = 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)),R5),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),T5))) )
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q,R5)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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(H4,As4,H8))))
              & relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ak(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H4),As4)),H4,H8)
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H4)),lim(product_unit,H8))) ) )
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% hoare_tripleI
tff(fact_454_hoare__tripleE,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H: heap_ext(product_unit),As: set(nat)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
       => ~ ! [R: A,H5: 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),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)),R),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),H5),T3)))
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),hoare_new_addrs(H,As,H5))))
               => ( relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ak(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H),As)),H,H5)
                 => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H)),lim(product_unit,H5))) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_455_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_al(A,fun(A,bool)),aTP_Lamp_am(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_456_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => ? [X3: A] :
            ( pp(aa(A,bool,P,X3))
            & ! [Y4: A] :
                ( pp(aa(A,bool,P,Y4))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X3))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_457_nat__descend__induct,axiom,
    ! [N2: nat,P: fun(nat,bool),M: nat] :
      ( ! [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),K2))
         => pp(aa(nat,bool,P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N2))
           => ( ! [I: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),I))
                 => pp(aa(nat,bool,P,I)) )
             => pp(aa(nat,bool,P,K2)) ) )
       => pp(aa(nat,bool,P,M)) ) ) ).

% nat_descend_induct
tff(fact_458_less__mono__imp__le__mono,axiom,
    ! [F3: fun(nat,nat),I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,I3)),aa(nat,nat,F3,J2))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F3,I2)),aa(nat,nat,F3,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_459_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( ( M != N2 )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% le_neq_implies_less
tff(fact_460_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
        | ( M = N2 ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% less_or_eq_imp_le
tff(fact_461_le__eq__less__or__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
        | ( M = N2 ) ) ) ).

% le_eq_less_or_eq
tff(fact_462_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% less_imp_le_nat
tff(fact_463_nat__less__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
        & ( M != N2 ) ) ) ).

% nat_less_le
tff(fact_464_subset__emptyI,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X3: A] : ~ pp(aa(set(A),bool,member(A,X3),A5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),bot_bot(set(A)))) ) ).

% subset_emptyI
tff(fact_465_preciseD,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn)),A3: A,P3: B,F4: assn,A4: A,F5: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( precise(A,B,R2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),R2,A3),P3)),F4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A4),P3)),F5))),H))
       => ( A3 = A4 ) ) ) ).

% preciseD
tff(fact_466_psubsetI,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( ( A5 != B4 )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4)) ) ) ).

% psubsetI
tff(fact_467_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
      ( ! [X3: A,Y3: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Y3))
         => pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Y3)) )
     => pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q)) ) ).

% predicate2I
tff(fact_468_sup2CI,axiom,
    ! [A: $tType,B: $tType,B4: fun(A,fun(B,bool)),X: A,Y: B,A5: fun(A,fun(B,bool))] :
      ( ( ~ pp(aa(B,bool,aa(A,fun(B,bool),B4,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y)) )
     => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A5),B4),X),Y)) ) ).

% sup2CI
tff(fact_469_predicate1I,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) )
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q)) ) ).

% predicate1I
tff(fact_470_inf1I,axiom,
    ! [A: $tType,A5: fun(A,bool),X: A,B4: fun(A,bool)] :
      ( pp(aa(A,bool,A5,X))
     => ( pp(aa(A,bool,B4,X))
       => pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A5),B4),X)) ) ) ).

% inf1I
tff(fact_471_inf2I,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),X: A,Y: B,B4: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),B4,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A5),B4),X),Y)) ) ) ).

% inf2I
tff(fact_472_sup1CI,axiom,
    ! [A: $tType,B4: fun(A,bool),X: A,A5: fun(A,bool)] :
      ( ( ~ pp(aa(A,bool,B4,X))
       => pp(aa(A,bool,A5,X)) )
     => pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A5),B4),X)) ) ).

% sup1CI
tff(fact_473_less__by__empty,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),B4: set(product_prod(A,A))] :
      ( ( A5 = bot_bot(set(product_prod(A,A))) )
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A5),B4)) ) ).

% less_by_empty
tff(fact_474_inf1E,axiom,
    ! [A: $tType,A5: fun(A,bool),B4: fun(A,bool),X: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A5),B4),X))
     => ~ ( pp(aa(A,bool,A5,X))
         => ~ pp(aa(A,bool,B4,X)) ) ) ).

% inf1E
tff(fact_475_inf2E,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),B4: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A5),B4),X),Y))
     => ~ ( pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y))
         => ~ pp(aa(B,bool,aa(A,fun(B,bool),B4,X),Y)) ) ) ).

% inf2E
tff(fact_476_sup1E,axiom,
    ! [A: $tType,A5: fun(A,bool),B4: fun(A,bool),X: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A5),B4),X))
     => ( ~ pp(aa(A,bool,A5,X))
       => pp(aa(A,bool,B4,X)) ) ) ).

% sup1E
tff(fact_477_sup2E,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),B4: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A5),B4),X),Y))
     => ( ~ pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),B4,X),Y)) ) ) ).

% sup2E
tff(fact_478_inf1D1,axiom,
    ! [A: $tType,A5: fun(A,bool),B4: fun(A,bool),X: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A5),B4),X))
     => pp(aa(A,bool,A5,X)) ) ).

% inf1D1
tff(fact_479_inf1D2,axiom,
    ! [A: $tType,A5: fun(A,bool),B4: fun(A,bool),X: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A5),B4),X))
     => pp(aa(A,bool,B4,X)) ) ).

% inf1D2
tff(fact_480_inf2D1,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),B4: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A5),B4),X),Y))
     => pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y)) ) ).

% inf2D1
tff(fact_481_inf2D2,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),B4: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A5),B4),X),Y))
     => pp(aa(B,bool,aa(A,fun(B,bool),B4,X),Y)) ) ).

% inf2D2
tff(fact_482_sup1I1,axiom,
    ! [A: $tType,A5: fun(A,bool),X: A,B4: fun(A,bool)] :
      ( pp(aa(A,bool,A5,X))
     => pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A5),B4),X)) ) ).

% sup1I1
tff(fact_483_sup1I2,axiom,
    ! [A: $tType,B4: fun(A,bool),X: A,A5: fun(A,bool)] :
      ( pp(aa(A,bool,B4,X))
     => pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A5),B4),X)) ) ).

% sup1I2
tff(fact_484_sup2I1,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),X: A,Y: B,B4: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y))
     => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A5),B4),X),Y)) ) ).

% sup2I1
tff(fact_485_sup2I2,axiom,
    ! [A: $tType,B: $tType,B4: fun(A,fun(B,bool)),X: A,Y: B,A5: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),B4,X),Y))
     => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A5),B4),X),Y)) ) ).

% sup2I2
tff(fact_486_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Y: B,Q: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
     => ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).

% rev_predicate2D
tff(fact_487_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,X))
     => ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
       => pp(aa(A,bool,Q,X)) ) ) ).

% rev_predicate1D
tff(fact_488_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).

% predicate2D
tff(fact_489_predicate1D,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),X: A] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
     => ( pp(aa(A,bool,P,X))
       => pp(aa(A,bool,Q,X)) ) ) ).

% predicate1D
tff(fact_490_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semila1105856199041335345_order(A,F3,Z2,Less_eq,Less)
     => ( ( Z2 = aa(A,A,aa(A,fun(A,A),F3,A3),B2) )
      <=> ( ( A3 = Z2 )
          & ( B2 = Z2 ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_491_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( semila1105856199041335345_order(A,F3,Z2,Less_eq,Less)
     => ( ( aa(A,A,aa(A,fun(A,A),F3,A3),B2) = Z2 )
      <=> ( ( A3 = Z2 )
          & ( B2 = Z2 ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_492_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X),Y)) ).

% bot2E
tff(fact_493_not__psubset__empty,axiom,
    ! [A: $tType,A5: set(A)] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),bot_bot(set(A)))) ).

% not_psubset_empty
tff(fact_494_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
        | ( A5 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_495_subset__psubset__trans,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B4),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),C4)) ) ) ).

% subset_psubset_trans
tff(fact_496_subset__not__subset__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
        & ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5)) ) ) ).

% subset_not_subset_eq
tff(fact_497_psubset__subset__trans,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),C4)) ) ) ).

% psubset_subset_trans
tff(fact_498_psubset__imp__subset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ).

% psubset_imp_subset
tff(fact_499_psubset__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
        & ( A5 != B4 ) ) ) ).

% psubset_eq
tff(fact_500_psubsetE,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5)) ) ) ).

% psubsetE
tff(fact_501_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F3),G3))
        <=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
            & ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G3),F3)) ) ) ) ).

% less_fun_def
tff(fact_502_less__assn__def,axiom,
    ! [A3: assn,B2: assn] :
      ( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less(assn),A3),B2))
    <=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less_eq(assn),A3),B2))
        & ( A3 != B2 ) ) ) ).

% less_assn_def
tff(fact_503_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P,Y4)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% measure_induct
tff(fact_504_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P,Y4)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% measure_induct_rule
tff(fact_505_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R3: A,S2: B,R2: set(product_prod(A,B)),S4: B] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R3),S2)),R2))
     => ( ( S4 = S2 )
       => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R3),S4)),R2)) ) ) ).

% ssubst_Pair_rhs
tff(fact_506_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M)) ) ) ).

% nat_neq_iff
tff(fact_507_less__not__refl,axiom,
    ! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N2)) ).

% less_not_refl
tff(fact_508_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
     => ( M != N2 ) ) ).

% less_not_refl2
tff(fact_509_less__not__refl3,axiom,
    ! [S2: nat,T6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T6))
     => ( S2 != T6 ) ) ).

% less_not_refl3
tff(fact_510_less__irrefl__nat,axiom,
    ! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N2)) ).

% less_irrefl_nat
tff(fact_511_nat__less__induct,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N5))
             => pp(aa(nat,bool,P,M4)) )
         => pp(aa(nat,bool,P,N5)) )
     => pp(aa(nat,bool,P,N2)) ) ).

% nat_less_induct
tff(fact_512_infinite__descent,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( ! [N5: nat] :
          ( ~ pp(aa(nat,bool,P,N5))
         => ? [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N5))
              & ~ pp(aa(nat,bool,P,M4)) ) )
     => pp(aa(nat,bool,P,N2)) ) ).

% infinite_descent
tff(fact_513_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_514_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,bool),V: fun(A,nat),X: A] :
      ( ! [X3: A] :
          ( ~ pp(aa(A,bool,P,X3))
         => ? [Y4: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X3)))
              & ~ pp(aa(A,bool,P,Y4)) ) )
     => pp(aa(A,bool,P,X)) ) ).

% infinite_descent_measure
tff(fact_515_le__refl,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N2)) ).

% le_refl
tff(fact_516_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K)) ) ) ).

% le_trans
tff(fact_517_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% eq_imp_le
tff(fact_518_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
       => ( M = N2 ) ) ) ).

% le_antisym
tff(fact_519_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M)) ) ).

% nat_le_linear
tff(fact_520_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => ? [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
            & ! [Y4: nat] :
                ( pp(aa(nat,bool,P,Y4))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_521_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ? [X3: A] :
          ( pp(aa(A,bool,P,X3))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P,Y4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,X3)),aa(A,nat,M,Y4))) ) ) ) ).

% ex_has_least_nat
tff(fact_522_sngr__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(A,ref(A),aTP_Lamp_an(A,fun(ref(A),assn))) ) ).

% sngr_prec
tff(fact_523_snga__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(list(A),array(A),aTP_Lamp_ao(list(A),fun(array(A),assn))) ) ).

% snga_prec
tff(fact_524_preciseD_H,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn)),A3: A,P3: B,F4: assn,H: product_prod(heap_ext(product_unit),set(nat)),A4: A,F5: assn] :
      ( precise(A,B,R2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A3),P3)),F4)),H))
       => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A4),P3)),F5)),H))
         => ( A3 = A4 ) ) ) ) ).

% preciseD'
tff(fact_525_prop__restrict,axiom,
    ! [A: $tType,X: A,Z4: set(A),X6: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,X),Z4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))))
       => pp(aa(A,bool,P,X)) ) ) ).

% prop_restrict
tff(fact_526_Collect__restrict,axiom,
    ! [A: $tType,X6: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))),X6)) ).

% Collect_restrict
tff(fact_527_precise__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn))] :
      ( precise(A,B,R2)
    <=> ! [A8: A,A9: A,H6: product_prod(heap_ext(product_unit),set(nat)),P4: B,F6: assn,F7: assn] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),R2,A8),P4)),F6)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A9),P4)),F7))),H6))
         => ( A8 = A9 ) ) ) ).

% precise_def
tff(fact_528_preciseI,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn))] :
      ( ! [A6: A,A10: A,H4: product_prod(heap_ext(product_unit),set(nat)),P5: B,F8: assn,F9: assn] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),R2,A6),P5)),F8)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A10),P5)),F9))),H4))
         => ( A6 = A10 ) )
     => precise(A,B,R2) ) ).

% preciseI
tff(fact_529_reflclp__idemp,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A))),fequal(A)) = aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)) ).

% reflclp_idemp
tff(fact_530_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T6),X5)) ) ) ).

% minf(8)
tff(fact_531_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),T6)) ) ) ).

% minf(6)
tff(fact_532_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T6),X5)) ) ) ).

% pinf(8)
tff(fact_533_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),T6)) ) ) ).

% pinf(6)
tff(fact_534_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B3: B,A4: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B3),A4))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A4),B3)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_535_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,P,A3))
           => ( ~ pp(aa(A,bool,P,B2))
             => ? [C3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
                  & ! [X5: A] :
                      ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X5))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),C3)) )
                     => pp(aa(A,bool,P,X5)) )
                  & ! [D5: A] :
                      ( ! [X3: A] :
                          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),D5)) )
                         => pp(aa(A,bool,P,X3)) )
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D5),C3)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_536_mod__h__bot__iff_I8_J,axiom,
    ! [C: $tType,R2: fun(C,assn),H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(C,R2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),bot_bot(set(nat)))))
    <=> ? [X4: C] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(C,assn,R2,X4)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),bot_bot(set(nat))))) ) ).

% mod_h_bot_iff(8)
tff(fact_537_bijective__Empty,axiom,
    ! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).

% bijective_Empty
tff(fact_538_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_539_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb))))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))))
               => ? [As13: set(nat),As23: set(nat)] :
                    ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As23) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As23) = bot_bot(set(nat)) )
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As13)))
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),As23))) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_540_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_ap(assn,fun(A,assn),C2)) = C2 ).

% ex_assn_const
tff(fact_541_mod__ex__dist,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(A,P)),H))
    <=> ? [X4: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P,X4)),H)) ) ).

% mod_ex_dist
tff(fact_542_psubsetD,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,member(A,C2),A5))
       => pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% psubsetD
tff(fact_543_less__set__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B4))) ) ).

% less_set_def
tff(fact_544_psubset__trans,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B4),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),C4)) ) ) ).

% psubset_trans
tff(fact_545_mod__exE,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(A,P)),H))
     => ~ ! [X3: A] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P,X3)),H)) ) ).

% mod_exE
tff(fact_546_mod__exI,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( ? [X5: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P,X5)),H))
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(A,P)),H)) ) ).

% mod_exI
tff(fact_547_ex__one__point__gen,axiom,
    ! [A: $tType,P: fun(A,assn),V2: A] :
      ( ! [H4: product_prod(heap_ext(product_unit),set(nat)),X3: A] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P,X3)),H4))
         => ( X3 = V2 ) )
     => ( ex_assn(A,P) = aa(A,assn,P,V2) ) ) ).

% ex_one_point_gen
tff(fact_548_ex__distrib__star,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_aq(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,P)),Q) ).

% ex_distrib_star
tff(fact_549_ex__join__or,axiom,
    ! [A: $tType,P: fun(A,assn),Q: fun(A,assn)] : ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ar(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_as(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) ).

% ex_join_or
tff(fact_550_ex__distrib__or,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_at(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,P)),Q) ).

% ex_distrib_or
tff(fact_551_ex__distrib__and,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_au(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,P)),Q) ).

% ex_distrib_and
tff(fact_552_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% verit_la_disequality
tff(fact_553_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).

% verit_comp_simplify1(2)
tff(fact_554_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A3: A] :
        ? [B5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B5))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B5),A3)) ) ) ).

% ex_gt_or_lt
tff(fact_555_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).

% verit_comp_simplify1(1)
tff(fact_556_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P6: fun(A,bool),Q: fun(A,bool),Q4: fun(A,bool)] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z5),X3))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P6,X3)) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z5),X3))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q4,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
               => ( ( pp(aa(A,bool,P,X5))
                    & pp(aa(A,bool,Q,X5)) )
                <=> ( pp(aa(A,bool,P6,X5))
                    & pp(aa(A,bool,Q4,X5)) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_557_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P6: fun(A,bool),Q: fun(A,bool),Q4: fun(A,bool)] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z5),X3))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P6,X3)) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z5),X3))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q4,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
               => ( ( pp(aa(A,bool,P,X5))
                    | pp(aa(A,bool,Q,X5)) )
                <=> ( pp(aa(A,bool,P6,X5))
                    | pp(aa(A,bool,Q4,X5)) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_558_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ( X5 != T6 ) ) ) ).

% pinf(3)
tff(fact_559_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ( X5 != T6 ) ) ) ).

% pinf(4)
tff(fact_560_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),T6)) ) ) ).

% pinf(5)
tff(fact_561_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T6),X5)) ) ) ).

% pinf(7)
tff(fact_562_pinf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X5: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),Z3),X5))
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_563_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P6: fun(A,bool),Q: fun(A,bool),Q4: fun(A,bool)] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z5))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P6,X3)) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z5))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q4,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
               => ( ( pp(aa(A,bool,P,X5))
                    & pp(aa(A,bool,Q,X5)) )
                <=> ( pp(aa(A,bool,P6,X5))
                    & pp(aa(A,bool,Q4,X5)) ) ) ) ) ) ) ).

% minf(1)
tff(fact_564_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P6: fun(A,bool),Q: fun(A,bool),Q4: fun(A,bool)] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z5))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P6,X3)) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z5))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q4,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
               => ( ( pp(aa(A,bool,P,X5))
                    | pp(aa(A,bool,Q,X5)) )
                <=> ( pp(aa(A,bool,P6,X5))
                    | pp(aa(A,bool,Q4,X5)) ) ) ) ) ) ) ).

% minf(2)
tff(fact_565_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ( X5 != T6 ) ) ) ).

% minf(3)
tff(fact_566_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ( X5 != T6 ) ) ) ).

% minf(4)
tff(fact_567_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),T6)) ) ) ).

% minf(5)
tff(fact_568_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T6: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T6),X5)) ) ) ).

% minf(7)
tff(fact_569_minf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X5: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),X5),Z3))
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_570_bijective__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( bijective(A,B,R2)
    <=> ( ! [X4: A,Y5: B,Z6: B] :
            ( ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y5)),R2))
              & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z6)),R2)) )
           => ( Y5 = Z6 ) )
        & ! [X4: A,Y5: A,Z6: B] :
            ( ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z6)),R2))
              & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y5),Z6)),R2)) )
           => ( X4 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_571_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb))
      <=> pp(Y) )
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb))))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( ( pp(Y)
                <=> ? [As12: set(nat),As22: set(nat)] :
                      ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As22) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As22) = bot_bot(set(nat)) )
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As12)))
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),As22))) ) )
               => ~ pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_572_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb))))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))))
               => ~ ? [As1: set(nat),As2: set(nat)] :
                      ( ( As4 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As2) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As2) = bot_bot(set(nat)) )
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As1)))
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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))) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_573_the__dflt__None__set,axiom,
    ! [A: $tType,X: set(A)] : the_default(set(A),bot_bot(set(A)),dflt_None_set(A,X)) = X ).

% the_dflt_None_set
tff(fact_574_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,bool)),R22: fun(A,fun(A,bool))] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R1),R22))
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),accp(A,R22)),accp(A,R1))) ) ).

% accp_subset
tff(fact_575_accp__subset__induct,axiom,
    ! [A: $tType,D4: fun(A,bool),R2: fun(A,fun(A,bool)),X: A,P: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),D4),accp(A,R2)))
     => ( ! [X3: A,Z3: A] :
            ( pp(aa(A,bool,D4,X3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,Z3),X3))
             => pp(aa(A,bool,D4,Z3)) ) )
       => ( pp(aa(A,bool,D4,X))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,D4,X3))
               => ( ! [Z5: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),R2,Z5),X3))
                     => pp(aa(A,bool,P,Z5)) )
                 => pp(aa(A,bool,P,X3)) ) )
           => pp(aa(A,bool,P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_576_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_av(A,fun(A,bool)),aTP_Lamp_aw(A,fun(A,bool)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_577_mod__h__bot__iff_I1_J,axiom,
    ! [B2: bool,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(pure_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)),H),bot_bot(set(nat)))))
    <=> pp(B2) ) ).

% mod_h_bot_iff(1)
tff(fact_578_uncurry__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,uncurry(B,C,A,F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F3,A3),B2) ).

% uncurry_apply
tff(fact_579_hoare__triple__effect,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H: heap_ext(product_unit),As: set(nat)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
       => ? [H5: heap_ext(product_unit),R: A,T3: nat] :
            ( heap_Time_effect(A,C2,H,H5,R,T3)
            & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),hoare_new_addrs(H,As,H5)))) ) ) ) ).

% hoare_triple_effect
tff(fact_580_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_ax(A,fun(A,bool)),aTP_Lamp_ay(A,fun(A,bool))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_581_wand__assnI,axiom,
    ! [H: heap_ext(product_unit),As: set(nat),Q: assn,R2: assn] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
     => ( ! [H5: heap_ext(product_unit),As6: set(nat)] :
            ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As6) = bot_bot(set(nat)) )
           => ( relH(As,H,H5)
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As)))
               => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6)))
                 => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(R2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6)))) ) ) ) )
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(wand_assn(Q,R2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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_assnI
tff(fact_582_pure__assn__eq__conv,axiom,
    ! [P: bool,Q: bool] :
      ( ( pure_assn(P) = pure_assn(Q) )
    <=> ( pp(P)
      <=> pp(Q) ) ) ).

% pure_assn_eq_conv
tff(fact_583_merge__pure__star,axiom,
    ! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fconj(A3,B2)) ).

% merge_pure_star
tff(fact_584_merge__pure__or,axiom,
    ! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fdisj(A3,B2)) ).

% merge_pure_or
tff(fact_585_pure__assn__eq__false__iff,axiom,
    ! [P: bool] :
      ( ( pure_assn(P) = bot_bot(assn) )
    <=> ~ pp(P) ) ).

% pure_assn_eq_false_iff
tff(fact_586_pure__false,axiom,
    pure_assn(fFalse) = bot_bot(assn) ).

% pure_false
tff(fact_587_merge__pure__and,axiom,
    ! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fconj(A3,B2)) ).

% merge_pure_and
tff(fact_588_in__range__empty,axiom,
    ! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),bot_bot(set(nat))))) ).

% in_range_empty
tff(fact_589_mod__pure__star__dist,axiom,
    ! [P: assn,B2: bool,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2))),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H))
        & pp(B2) ) ) ).

% mod_pure_star_dist
tff(fact_590_in__range__dist__union,axiom,
    ! [H: heap_ext(product_unit),As: set(nat),As3: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As3))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As3))) ) ) ).

% in_range_dist_union
tff(fact_591_precise__extr__pure_I1_J,axiom,
    ! [B: $tType,A: $tType,P: bool,R2: fun(A,fun(B,assn))] :
      ( precise(A,B,aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_az(bool,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),P),R2))
    <=> ( pp(P)
       => precise(A,B,R2) ) ) ).

% precise_extr_pure(1)
tff(fact_592_precise__extr__pure_I2_J,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn)),P: bool] :
      ( precise(A,B,aa(bool,fun(A,fun(B,assn)),aTP_Lamp_ba(fun(A,fun(B,assn)),fun(bool,fun(A,fun(B,assn))),R2),P))
    <=> ( pp(P)
       => precise(A,B,R2) ) ) ).

% precise_extr_pure(2)
tff(fact_593_effect__LetI,axiom,
    ! [B: $tType,A: $tType,X: A,T6: A,F3: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: B,N2: nat] :
      ( ( X = T6 )
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),F3,X),H,H3,R3,N2)
       => heap_Time_effect(B,aa(A,heap_Time_Heap(B),F3,T6),H,H3,R3,N2) ) ) ).

% effect_LetI
tff(fact_594_ordering__top_Oextremum__uniqueI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
       => ( A3 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_595_ordering__top_Onot__eq__extremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( ( A3 != Top )
      <=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),Top)) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_596_ordering__top_Oextremum__unique,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
      <=> ( A3 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_597_ordering__top_Oextremum__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Top),A3)) ) ).

% ordering_top.extremum_strict
tff(fact_598_ordering__top_Oextremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),Top)) ) ).

% ordering_top.extremum
tff(fact_599_models__in__range,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H))
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H)) ) ).

% models_in_range
tff(fact_600_relH__in__rangeI_I2_J,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(As,H,H3)
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))) ) ).

% relH_in_rangeI(2)
tff(fact_601_relH__in__rangeI_I1_J,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( relH(As,H,H3)
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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))) ) ).

% relH_in_rangeI(1)
tff(fact_602_relH__refl,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
     => relH(As,H,H) ) ).

% relH_refl
tff(fact_603_in__range__subset,axiom,
    ! [As: set(nat),As3: set(nat),H: heap_ext(product_unit)] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As),As3))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As3)))
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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))) ) ) ).

% in_range_subset
tff(fact_604_effectI,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),R3: A,H3: heap_ext(product_unit),N2: 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)),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),H3),N2))) )
     => heap_Time_effect(A,C2,H,H3,R3,N2) ) ).

% effectI
tff(fact_605_effect__def,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,N2: nat] :
      ( heap_Time_effect(A,C2,H,H3,R3,N2)
    <=> ( 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)),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),H3),N2))) ) ) ).

% effect_def
tff(fact_606_in__range_Oelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ! [X3: nat] :
                ( pp(aa(set(nat),bool,member(nat,X3),As4))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H4))) ) ) ) ).

% in_range.elims(3)
tff(fact_607_in__range_Oelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ~ ! [X5: nat] :
                  ( pp(aa(set(nat),bool,member(nat,X5),As4))
                 => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),lim(product_unit,H4))) ) ) ) ).

% in_range.elims(2)
tff(fact_608_in__range_Oelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
      <=> pp(Y) )
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ( pp(Y)
            <=> ~ ! [X4: nat] :
                    ( pp(aa(set(nat),bool,member(nat,X4),As4))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H4))) ) ) ) ) ).

% in_range.elims(1)
tff(fact_609_in__range_Osimps,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),As))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H))) ) ) ).

% in_range.simps
tff(fact_610_Heap__eqI,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),G3: heap_Time_Heap(A)] :
      ( ! [H4: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H4) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,G3),H4)
     => ( F3 = G3 ) ) ).

% Heap_eqI
tff(fact_611_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_612_the__default_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: A,X: A] : the_default(A,Uu,aa(A,option(A),some(A),X)) = X ).

% the_default.simps(1)
tff(fact_613_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
              & ! [H5: heap_ext(product_unit),As6: set(nat)] :
                  ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As6) = bot_bot(set(nat)) )
                    & relH(As4,H4,H5)
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As4)))
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6))) )
                 => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As6)))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_614_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
                & ! [H8: heap_ext(product_unit),As7: set(nat)] :
                    ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As7) = bot_bot(set(nat)) )
                      & relH(As4,H4,H8)
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As4)))
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H8),As7))) )
                   => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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)),As4),As7)))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_615_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb))
      <=> pp(Y) )
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ( pp(Y)
            <=> ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
                  & ! [H7: heap_ext(product_unit),As8: set(nat)] :
                      ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As8) = bot_bot(set(nat)) )
                        & relH(As4,H4,H7)
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As4)))
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As8))) )
                     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As8)))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_616_wand__raw_Osimps,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
        & ! [H7: heap_ext(product_unit),As8: set(nat)] :
            ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As8) = bot_bot(set(nat)) )
              & relH(As,H,H7)
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As)))
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As8))) )
           => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As8)))) ) ) ) ).

% wand_raw.simps
tff(fact_617_in__measure,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measure(A,F3)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y))) ) ).

% in_measure
tff(fact_618_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb))))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))))
               => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
                  & ! [H5: heap_ext(product_unit),As6: set(nat)] :
                      ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As6) = bot_bot(set(nat)) )
                        & relH(As4,H4,H5)
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As4)))
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6))) )
                     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As6)))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_619_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb))))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))))
               => ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
                    & ! [H8: heap_ext(product_unit),As7: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As7) = bot_bot(set(nat)) )
                          & relH(As4,H4,H8)
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As4)))
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H8),As7))) )
                       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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)),As4),As7)))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_620_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb))
      <=> pp(Y) )
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb))))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( ( pp(Y)
                <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
                    & ! [H7: heap_ext(product_unit),As8: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As4),As8) = bot_bot(set(nat)) )
                          & relH(As4,H4,H7)
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As4)))
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As8))) )
                       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As4),As8)))) ) ) )
               => ~ pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_621_in__range_Opelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
               => ! [X3: nat] :
                    ( pp(aa(set(nat),bool,member(nat,X3),As4))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H4))) ) ) ) ) ) ).

% in_range.pelims(3)
tff(fact_622_in__range_Opelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
               => ~ ! [X5: nat] :
                      ( pp(aa(set(nat),bool,member(nat,X5),As4))
                     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),lim(product_unit,H4))) ) ) ) ) ) ).

% in_range.pelims(2)
tff(fact_623_in__range_Opelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
      <=> pp(Y) )
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( ( pp(Y)
                <=> ! [X4: nat] :
                      ( pp(aa(set(nat),bool,member(nat,X4),As4))
                     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H4))) ) )
               => ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))) ) ) ) ) ).

% in_range.pelims(1)
tff(fact_624_wand__assn__def,axiom,
    ! [P: assn,Q: assn] : wand_assn(P,Q) = abs_assn(wand_raw(rep_assn(P),rep_assn(Q))) ).

% wand_assn_def
tff(fact_625_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aTP_Lamp_bb(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P))) ).

% eq_subset
tff(fact_626_sup__option__def,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X),Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_bd(option(A),fun(option(A),fun(A,option(A))),X),Y),X) ) ).

% sup_option_def
tff(fact_627_dflt__None__set__def,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( ( S = bot_bot(set(A)) )
       => ( dflt_None_set(A,S) = none(set(A)) ) )
      & ( ( S != bot_bot(set(A)) )
       => ( dflt_None_set(A,S) = aa(set(A),option(set(A)),some(set(A)),S) ) ) ) ).

% dflt_None_set_def
tff(fact_628_times__assn__def,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = abs_assn(times_assn_raw(rep_assn(P),rep_assn(Q))) ).

% times_assn_def
tff(fact_629_Set_Ois__empty__def,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_empty(A,A5)
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% Set.is_empty_def
tff(fact_630_hoare__triple__success,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H: heap_ext(product_unit),As: set(nat)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
       => heap_Time_success(A,C2,H) ) ) ).

% hoare_triple_success
tff(fact_631_subset__Collect__iff,axiom,
    ! [A: $tType,B4: set(A),A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),A5),P))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B4))
           => pp(aa(A,bool,P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_632_subset__CollectI,axiom,
    ! [A: $tType,B4: set(A),A5: set(A),Q: fun(A,bool),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),B4))
           => ( pp(aa(A,bool,Q,X3))
             => pp(aa(A,bool,P,X3)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),B4),Q))),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),A5),P)))) ) ) ).

% subset_CollectI
tff(fact_633_conj__subset__def,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ag(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(fun(A,bool),set(A),collect(A),P)))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(fun(A,bool),set(A),collect(A),Q))) ) ) ).

% conj_subset_def
tff(fact_634_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y5: A] : X = aa(A,option(A),some(A),Y5) ) ).

% not_None_eq
tff(fact_635_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y5: A] : X != aa(A,option(A),some(A),Y5)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_636_Rep__assn__inverse,axiom,
    ! [X: assn] : abs_assn(rep_assn(X)) = X ).

% Rep_assn_inverse
tff(fact_637_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),none(A)),X)) ) ).

% less_eq_option_None_code
tff(fact_638_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : ~ pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),X),none(A))) ) ).

% less_option_None
tff(fact_639_sup__None__2,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X),none(A)) = X ) ).

% sup_None_2
tff(fact_640_sup__None__1,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),none(A)),Y) = Y ) ).

% sup_None_1
tff(fact_641_inf__None__2,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),none(A)) = none(A) ) ).

% inf_None_2
tff(fact_642_inf__None__1,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),none(A)),Y) = none(A) ) ).

% inf_None_1
tff(fact_643_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),X)),none(A))) ) ).

% less_eq_option_Some_None
tff(fact_644_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),aa(A,option(A),some(A),X))) ) ).

% less_option_None_Some_code
tff(fact_645_the__dflt__None__empty,axiom,
    ! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).

% the_dflt_None_empty
tff(fact_646_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: fun(A,B)] : case_option(B,A,F1,F22,none(A)) = F1 ).

% option.simps(4)
tff(fact_647_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_648_success__LetI,axiom,
    ! [A: $tType,B: $tType,X: A,T6: A,F3: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
      ( ( X = T6 )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),F3,X),H)
       => heap_Time_success(B,aa(A,heap_Time_Heap(B),F3,T6),H) ) ) ).

% success_LetI
tff(fact_649_option_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,B),Option: option(A)] : aa(B,C,H,case_option(B,A,F1,F22,Option)) = case_option(C,A,aa(B,C,H,F1),aa(fun(A,B),fun(A,C),aTP_Lamp_be(fun(B,C),fun(fun(A,B),fun(A,C)),H),F22),Option) ).

% option.case_distrib
tff(fact_650_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X2: A] : none(A) != aa(A,option(A),some(A),X2) ).

% option.distinct(1)
tff(fact_651_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X2: A] :
      ( ( Option = aa(A,option(A),some(A),X2) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_652_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X22: A] : Y != aa(A,option(A),some(A),X22) ) ).

% option.exhaust
tff(fact_653_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ? [X_12: option(A)] : pp(aa(option(A),bool,P,X_12))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        | ? [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_ex
tff(fact_654_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ! [X_12: option(A)] : pp(aa(option(A),bool,P,X_12))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        & ! [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_all
tff(fact_655_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P: fun(option(A),fun(option(B),bool)),Y: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
     => ( ( ( Y = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
       => ( ! [A6: A,B5: B] :
              ( ( X = aa(A,option(A),some(A),A6) )
             => ( ( Y = aa(B,option(B),some(B),B5) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_656_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),none(A)),X)) ) ).

% less_eq_option_None
tff(fact_657_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),X),none(A)))
         => ( X = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_658_Abs__assn__eqI_I1_J,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Pr: assn] :
      ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,H4))
        <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Pr),H4)) )
     => ( abs_assn(P) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_659_Abs__assn__eqI_I2_J,axiom,
    ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Pr: assn] :
      ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,H4))
        <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Pr),H4)) )
     => ( Pr = abs_assn(P) ) ) ).

% Abs_assn_eqI(2)
tff(fact_660_option_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: fun(A,B),X2: A] : case_option(B,A,F1,F22,aa(A,option(A),some(A),X2)) = aa(A,B,F22,X2) ).

% option.simps(5)
tff(fact_661_the__default_Osimps_I2_J,axiom,
    ! [A: $tType,X: A] : the_default(A,X,none(A)) = X ).

% the_default.simps(2)
tff(fact_662_inf__option__def,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),Y) = case_option(option(A),A,none(A),aTP_Lamp_bg(option(A),fun(A,option(A)),Y),X) ) ).

% inf_option_def
tff(fact_663_bot__assn__def,axiom,
    bot_bot(assn) = abs_assn(aTP_Lamp_bh(product_prod(heap_ext(product_unit),set(nat)),bool)) ).

% bot_assn_def
tff(fact_664_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),X))
         => ? [Z3: A] : X = aa(A,option(A),some(A),Z3) ) ) ).

% less_option_None_is_Some
tff(fact_665_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),aa(A,option(A),some(A),X))) ) ).

% less_option_None_Some
tff(fact_666_sup__assn__def,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q) = abs_assn(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_bi(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q)) ).

% sup_assn_def
tff(fact_667_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),R2: bool,X: A,Y: B] :
      ( ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q))
        & pp(R2) )
     => ( pp(R2)
        & ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
         => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ) ).

% predicate2D_conj
tff(fact_668_inf__assn__def,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q) = abs_assn(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_bj(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q)) ).

% inf_assn_def
tff(fact_669_successE,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F3,H)
     => ~ ! [R: A,H5: 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,F3),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)),R),H5)) ) ).

% successE
tff(fact_670_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : rel_of(A,B,aTP_Lamp_bk(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_671_success__bind__executeI,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H3: heap_ext(product_unit),N2: nat,G3: 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,F3),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)),X),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),H3),N2))) )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G3,X),H3)
       => heap_Time_success(B,heap_Time_bind(A,B,F3,G3),H) ) ) ).

% success_bind_executeI
tff(fact_672_sngr__assn__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),X: A] : sngr_assn(A,R3,X) = abs_assn(sngr_assn_raw(A,R3,X)) ) ).

% sngr_assn_def
tff(fact_673_snga__assn__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: array(A),A3: list(A)] : snga_assn(A,R3,A3) = abs_assn(snga_assn_raw(A,R3,A3)) ) ).

% snga_assn_def
tff(fact_674_set__to__map__empty,axiom,
    ! [A: $tType,B: $tType,X5: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X5) = none(B) ).

% set_to_map_empty
tff(fact_675_map__to__set__empty,axiom,
    ! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_bk(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% map_to_set_empty
tff(fact_676_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: bool,Q: fun(A,bool),X: option(A),R2: fun(B,bool),F3: B,G3: fun(A,B)] :
      ( pp(case_option(bool,A,P,Q,X))
     => ( ( pp(P)
         => pp(aa(B,bool,R2,F3)) )
       => ( ! [Q5: A] :
              ( pp(aa(A,bool,Q,Q5))
             => pp(aa(B,bool,R2,aa(A,B,G3,Q5))) )
         => pp(aa(B,bool,R2,case_option(B,A,F3,G3,X))) ) ) ) ).

% disjE_realizer2
tff(fact_677_pure__assn__def,axiom,
    ! [B2: bool] : pure_assn(B2) = abs_assn(pure_assn_raw(heap_ext(product_unit),nat,B2)) ).

% pure_assn_def
tff(fact_678_set__to__map__empty__iff_I2_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ( aTP_Lamp_bk(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_679_internal__case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc5280177257484947105e_prod(B,C,A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),C2,A3),B2) ).

% internal_case_prod_conv
tff(fact_680_Heap__Time__Monad_Obind__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: heap_Time_Heap(A),G3: fun(A,heap_Time_Heap(C)),K: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,heap_Time_bind(A,C,F3,G3),K) = heap_Time_bind(A,B,F3,aa(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aTP_Lamp_bl(fun(A,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),G3),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_681_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V2: option(product_prod(A,B))] :
      ( ! [X4: A,Y5: 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),X4),Y5))
    <=> ( V2 = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_682_map__to__set__inverse,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : set_to_map(A,B,map_to_set(A,B,M)) = M ).

% map_to_set_inverse
tff(fact_683_execute__bind_I2_J,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),G3: 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,F3),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
     => ( 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,F3,G3)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_bind(2)
tff(fact_684_distrib__if__bind,axiom,
    ! [A: $tType,B: $tType,B2: bool,C2: heap_Time_Heap(B),D3: heap_Time_Heap(B),F3: fun(B,heap_Time_Heap(A))] :
      ( ( pp(B2)
       => ( heap_Time_bind(B,A,if(heap_Time_Heap(B),B2,C2,D3),F3) = heap_Time_bind(B,A,C2,F3) ) )
      & ( ~ pp(B2)
       => ( heap_Time_bind(B,A,if(heap_Time_Heap(B),B2,C2,D3),F3) = heap_Time_bind(B,A,D3,F3) ) ) ) ).

% distrib_if_bind
tff(fact_685_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(case_option(bool,A,fFalse,aTP_Lamp_bm(A,bool),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_686_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(case_option(bool,A,fTrue,aTP_Lamp_ae(A,bool),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_687_success__def,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F3,H)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% success_def
tff(fact_688_successI,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
     => heap_Time_success(A,F3,H) ) ).

% successI
tff(fact_689_case__optionE,axiom,
    ! [A: $tType,P: bool,Q: fun(A,bool),X: option(A)] :
      ( pp(case_option(bool,A,P,Q,X))
     => ( ( ( X = none(A) )
         => ~ pp(P) )
       => ~ ! [Y3: A] :
              ( ( X = aa(A,option(A),some(A),Y3) )
             => ~ pp(aa(A,bool,Q,Y3)) ) ) ) ).

% case_optionE
tff(fact_690_map__to__set__empty__iff_I1_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( map_to_set(A,B,M) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).

% map_to_set_empty_iff(1)
tff(fact_691_map__to__set__empty__iff_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M) )
    <=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).

% map_to_set_empty_iff(2)
tff(fact_692_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),X),Y))
        <=> pp(case_option(bool,A,fTrue,aTP_Lamp_bn(option(A),fun(A,bool),Y),X)) ) ) ).

% less_eq_option_def
tff(fact_693_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),X),Y))
        <=> pp(case_option(bool,A,fFalse,aTP_Lamp_bp(option(A),fun(A,bool),X),Y)) ) ) ).

% less_option_def
tff(fact_694_Heap__cases,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( ! [X3: A,H5: 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,F3),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)),X3),H5))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% Heap_cases
tff(fact_695_timeFrame_Ocases,axiom,
    ! [A: $tType,X: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( ! [N5: nat,R: A,H4: heap_ext(product_unit),N4: nat] : X != 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)),R),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),N4))))
     => ~ ! [N5: nat] : X != 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_696_set__to__map__empty__iff_I1_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ! [X4: A] : aa(A,option(B),set_to_map(A,B,S),X4) = none(B)
    <=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(1)
tff(fact_697_pure__assn__raw_Oelims_I3_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( ~ pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ~ ! [H4: A,As4: set(B)] :
            ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4) )
           => ( ( As4 = bot_bot(set(B)) )
              & pp(X) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_698_pure__assn__raw_Oelims_I2_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ~ ! [H4: A,As4: set(B)] :
            ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4) )
           => ~ ( ( As4 = bot_bot(set(B)) )
                & pp(X) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_699_pure__assn__raw_Oelims_I1_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B)),Y: bool] :
      ( ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
      <=> pp(Y) )
     => ~ ! [H4: A,As4: set(B)] :
            ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4) )
           => ( pp(Y)
            <=> ~ ( ( As4 = bot_bot(set(B)) )
                  & pp(X) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_700_pure__assn__raw_Osimps,axiom,
    ! [B: $tType,A: $tType,B2: bool,H: A,As: set(B)] :
      ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,B2),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)) )
        & pp(B2) ) ) ).

% pure_assn_raw.simps
tff(fact_701_pure__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType,B: $tType,X: bool,Xa2: product_prod(A,set(B)),Y: bool] :
      ( ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
      <=> pp(Y) )
     => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa2)))
       => ~ ! [H4: A,As4: set(B)] :
              ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4) )
             => ( ( pp(Y)
                <=> ( ( As4 = bot_bot(set(B)) )
                    & pp(X) ) )
               => ~ pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4)))) ) ) ) ) ).

% pure_assn_raw.pelims(1)
tff(fact_702_pure__assn__raw_Opelims_I2_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa2)))
       => ~ ! [H4: A,As4: set(B)] :
              ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4) )
             => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4))))
               => ~ ( ( As4 = bot_bot(set(B)) )
                    & pp(X) ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
tff(fact_703_pure__assn__raw_Opelims_I3_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( ~ pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa2)))
       => ~ ! [H4: A,As4: set(B)] :
              ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4) )
             => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H4),As4))))
               => ( ( As4 = bot_bot(set(B)) )
                  & pp(X) ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
tff(fact_704_execute__bind_I1_J,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H3: heap_ext(product_unit),N2: nat,G3: 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,F3),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)),X),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),H3),N2))) )
     => ( 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,F3,G3)),H) = heap_Time_timeFrame(B,N2,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),G3,X)),H3)) ) ) ).

% execute_bind(1)
tff(fact_705_execute__bind__eq__SomeI,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H3: heap_ext(product_unit),N2: nat,G3: fun(A,heap_Time_Heap(B)),Y: B,H9: 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,F3),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)),X),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),H3),N2))) )
     => ( ( 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),G3,X)),H3) = 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),H9),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,F3,G3)),H) = 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),H9),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_706_set__to__map__def,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(B,A)),K: B] : aa(B,option(A),set_to_map(B,A,S),K) = eps_Opt(A,aa(B,fun(A,bool),aTP_Lamp_bq(set(product_prod(B,A)),fun(B,fun(A,bool)),S),K)) ).

% set_to_map_def
tff(fact_707_execute__guard_I2_J,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( pp(aa(heap_ext(product_unit),bool,P,H))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_guard(A,P,F3)),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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H)) ) ) ).

% execute_guard(2)
tff(fact_708_curry__uncurry__id,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : product_curry(A,B,C,uncurry(A,B,C,F3)) = F3 ).

% curry_uncurry_id
tff(fact_709_uncurry__curry__id,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : uncurry(A,B,C,product_curry(A,B,C,F3)) = F3 ).

% uncurry_curry_id
tff(fact_710_mlex__leq,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2))) ) ) ).

% mlex_leq
tff(fact_711_curryI,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F3),A3),B2)) ) ).

% curryI
tff(fact_712_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% nat_add_left_cancel_less
tff(fact_713_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% nat_add_left_cancel_le
tff(fact_714_curry__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(product_prod(B,C),A),A3: B,B2: C] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F3),A3),B2) = aa(product_prod(B,C),A,F3,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) ).

% curry_conv
tff(fact_715_some__opt__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : eps_Opt(A,aa(A,fun(A,bool),fequal(A),X)) = aa(A,option(A),some(A),X) ).

% some_opt_sym_eq_trivial
tff(fact_716_some__opt__eq__trivial,axiom,
    ! [A: $tType,X: A] : eps_Opt(A,aTP_Lamp_br(A,fun(A,bool),X)) = aa(A,option(A),some(A),X) ).

% some_opt_eq_trivial
tff(fact_717_some__opt__false__trivial,axiom,
    ! [A: $tType] : eps_Opt(A,aTP_Lamp_ae(A,bool)) = none(A) ).

% some_opt_false_trivial
tff(fact_718_curry__K,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: C,X5: A,Xa3: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,aTP_Lamp_bs(C,fun(product_prod(A,B),C),C2)),X5),Xa3) = C2 ).

% curry_K
tff(fact_719_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K)) ) ).

% add_lessD1
tff(fact_720_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_less_mono
tff(fact_721_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),I2)) ).

% not_add_less1
tff(fact_722_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),I2)) ).

% not_add_less2
tff(fact_723_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_less_mono1
tff(fact_724_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_less_add1
tff(fact_725_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_less_add2
tff(fact_726_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% less_add_eq_less
tff(fact_727_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N2))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) ) ) ).

% add_leE
tff(fact_728_le__add1,axiom,
    ! [N2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M))) ).

% le_add1
tff(fact_729_le__add2,axiom,
    ! [N2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2))) ).

% le_add2
tff(fact_730_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% add_leD1
tff(fact_731_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) ) ).

% add_leD2
tff(fact_732_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
     => ? [N5: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N5) ) ).

% le_Suc_ex
tff(fact_733_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_le_mono
tff(fact_734_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_le_mono1
tff(fact_735_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_le_add1
tff(fact_736_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_le_add2
tff(fact_737_nat__le__iff__add,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
    <=> ? [K3: nat] : N2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_738_curryE,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F3),A3),B2))
     => pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryE
tff(fact_739_curryD,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F3),A3),B2))
     => pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryD
tff(fact_740_timeFrame_Osimps_I1_J,axiom,
    ! [A: $tType,N2: nat,R3: A,H: heap_ext(product_unit),N6: nat] : heap_Time_timeFrame(A,N2,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),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)),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),N2),N6)))) ).

% timeFrame.simps(1)
tff(fact_741_timeFrame_Oelims,axiom,
    ! [A: $tType,X: nat,Xa2: 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,X,Xa2) = Y )
     => ( ! [R: A,H4: heap_ext(product_unit),N4: nat] :
            ( ( Xa2 = 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)),R),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),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)),R),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),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N4)))) ) )
       => ~ ( ( Xa2 = 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_742_mono__nat__linear__lb,axiom,
    ! [F3: fun(nat,nat),M: nat,K: nat] :
      ( ! [M5: nat,N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,M5)),aa(nat,nat,F3,N5))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F3,M)),K)),aa(nat,nat,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_743_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F3: fun(A,nat),N2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ? [Y4: A] :
                ( pp(aa(A,bool,P,Y4))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X3))) ) )
       => ? [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,K)),N2))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_744_Eps__Opt__eq__Some,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ! [X7: A] :
          ( pp(aa(A,bool,P,X))
         => ( pp(aa(A,bool,P,X7))
           => ( X7 = X ) ) )
     => ( ( eps_Opt(A,P) = aa(A,option(A),some(A),X) )
      <=> pp(aa(A,bool,P,X)) ) ) ).

% Eps_Opt_eq_Some
tff(fact_745_Eps__Opt__eq__Some__implies,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ( eps_Opt(A,P) = aa(A,option(A),some(A),X) )
     => pp(aa(A,bool,P,X)) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_746_execute__guard_I1_J,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ~ pp(aa(heap_ext(product_unit),bool,P,H))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_guard(A,P,F3)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_guard(1)
tff(fact_747_curry__def,axiom,
    ! [C: $tType,A: $tType,B: $tType,X5: fun(product_prod(A,B),C),Xa3: A,Xb2: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X5),Xa3),Xb2) = aa(product_prod(A,B),C,X5,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa3),Xb2)) ).

% curry_def
tff(fact_748_mlex__less,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2))) ) ).

% mlex_less
tff(fact_749_mlex__iff,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
        | ( ( aa(A,nat,F3,X) = aa(A,nat,F3,Y) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ).

% mlex_iff
tff(fact_750_timeFrame_Opelims,axiom,
    ! [A: $tType,X: nat,Xa2: 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,X,Xa2) = Y )
     => ( pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,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)))),X),Xa2)))
       => ( ! [R: A,H4: heap_ext(product_unit),N4: nat] :
              ( ( Xa2 = 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)),R),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),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)),R),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),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N4)))) )
               => ~ pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,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)))),X),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)),R),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),N4)))))) ) )
         => ~ ( ( Xa2 = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
             => ( ( Y = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
               => ~ pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,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)))),X),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_751_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_left
tff(fact_752_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_right
tff(fact_753_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_left
tff(fact_754_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_right
tff(fact_755_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_756_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_757_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_le_less_mono
tff(fact_758_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_less_le_mono
tff(fact_759_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( ( A3 != B2 )
            & ( C2 != D3 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_760_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ).

% mult_le_mono2
tff(fact_761_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ).

% mult_le_mono1
tff(fact_762_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L))) ) ) ).

% mult_le_mono
tff(fact_763_le__square,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_square
tff(fact_764_le__cube,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)))) ).

% le_cube
tff(fact_765_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K)) ).

% add_mult_distrib
tff(fact_766_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) ).

% add_mult_distrib2
tff(fact_767_mlex__bound,axiom,
    ! [A3: nat,A5: nat,B2: nat,N7: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N7))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N7))) ) ) ).

% mlex_bound
tff(fact_768_mlex__fst__decrI,axiom,
    ! [A3: nat,A4: nat,B2: nat,N7: nat,B3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N7))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B3),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N7)),B3))) ) ) ) ).

% mlex_fst_decrI
tff(fact_769_mlex__snd__decrI,axiom,
    ! [A3: nat,A4: nat,B2: nat,B3: nat,N7: nat] :
      ( ( A3 = A4 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N7)),B3))) ) ) ).

% mlex_snd_decrI
tff(fact_770_mlex__leI,axiom,
    ! [A3: nat,A4: nat,B2: nat,B3: nat,N7: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N7)),B3))) ) ) ).

% mlex_leI
tff(fact_771_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_772_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_773_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_774_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_775_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_776_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ? [C5: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C5) ) ) ).

% le_iff_add
tff(fact_777_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_right_mono
tff(fact_778_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ~ ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ) ).

% less_eqE
tff(fact_779_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_left_mono
tff(fact_780_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_mono
tff(fact_781_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_782_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_783_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_784_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_785_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_786_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_strict_right_mono
tff(fact_787_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_strict_left_mono
tff(fact_788_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_strict_mono
tff(fact_789_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_790_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_791_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_792_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W2: A,Y: A,X: A,Z2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
        <=> ( ( W2 = X )
            | ( Y = Z2 ) ) ) ) ).

% crossproduct_eq
tff(fact_793_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_794_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_795_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_796_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% distrib_left
tff(fact_797_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_798_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E3)),C2) ) ).

% combine_common_factor
tff(fact_799_execute__raise,axiom,
    ! [A: $tType,S2: list(char),X5: 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_raise(A,S2)),X5) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% execute_raise
tff(fact_800_guard__def,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_guard(A,P,F3) = heap_Time_Heap2(A,aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_bt(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),P),F3)) ).

% guard_def
tff(fact_801_execute__assert_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,H: heap_ext(product_unit)] :
      ( ~ pp(aa(A,bool,P,X))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P,X)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_assert(2)
tff(fact_802_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: bool] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),pure_assn(B2)))
    <=> ( ! [H6: product_prod(heap_ext(product_unit),set(nat))] :
            ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H6))
           => pp(B2) )
        & entails(P,Q) ) ) ).

% ent_pure_post_iff
tff(fact_803_Heap__execute,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A)] : heap_Time_Heap2(A,heap_Time_execute(A,F3)) = F3 ).

% Heap_execute
tff(fact_804_triv__exI,axiom,
    ! [A: $tType,Q: fun(A,assn),X: A] : entails(aa(A,assn,Q,X),ex_assn(A,Q)) ).

% triv_exI
tff(fact_805_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: bool,Q: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2)),Q)
    <=> ( pp(B2)
       => entails(P,Q) ) ) ).

% ent_pure_pre_iff
tff(fact_806_ent__false__iff,axiom,
    ! [P: assn] :
      ( entails(P,bot_bot(assn))
    <=> ! [H6: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H6)) ) ).

% ent_false_iff
tff(fact_807_ent__trans,axiom,
    ! [P: assn,Q: assn,R2: assn] :
      ( entails(P,Q)
     => ( entails(Q,R2)
       => entails(P,R2) ) ) ).

% ent_trans
tff(fact_808_ent__refl,axiom,
    ! [P: assn] : entails(P,P) ).

% ent_refl
tff(fact_809_ent__iffI,axiom,
    ! [A5: assn,B4: assn] :
      ( entails(A5,B4)
     => ( entails(B4,A5)
       => ( A5 = B4 ) ) ) ).

% ent_iffI
tff(fact_810_raise__def,axiom,
    ! [A: $tType,S2: list(char)] : heap_Time_raise(A,S2) = heap_Time_Heap2(A,aTP_Lamp_bu(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).

% raise_def
tff(fact_811_entails__def,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
    <=> ! [H6: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H6))
         => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),H6)) ) ) ).

% entails_def
tff(fact_812_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H4))
         => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),H4)) )
     => entails(P,Q) ) ).

% entailsI
tff(fact_813_entailsD,axiom,
    ! [P: assn,Q: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( entails(P,Q)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H))
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),H)) ) ) ).

% entailsD
tff(fact_814_ent__fwd,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat)),Q: assn] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H))
     => ( entails(P,Q)
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q),H)) ) ) ).

% ent_fwd
tff(fact_815_ent__star__mono,axiom,
    ! [P: assn,P6: assn,Q: assn,Q4: assn] :
      ( entails(P,P6)
     => ( entails(Q,Q4)
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P6),Q4)) ) ) ).

% ent_star_mono
tff(fact_816_ent__disjE,axiom,
    ! [A5: assn,C4: assn,B4: assn] :
      ( entails(A5,C4)
     => ( entails(B4,C4)
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B4),C4) ) ) ).

% ent_disjE
tff(fact_817_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),R2)
     => entails(P,R2) ) ).

% ent_disjI1
tff(fact_818_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),R2)
     => entails(Q,R2) ) ).

% ent_disjI2
tff(fact_819_ent__disjI1_H,axiom,
    ! [A5: assn,B4: assn,C4: assn] :
      ( entails(A5,B4)
     => entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B4),C4)) ) ).

% ent_disjI1'
tff(fact_820_ent__disjI2_H,axiom,
    ! [A5: assn,C4: assn,B4: assn] :
      ( entails(A5,C4)
     => entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B4),C4)) ) ).

% ent_disjI2'
tff(fact_821_ent__disjI1__direct,axiom,
    ! [A5: assn,B4: assn] : entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B4)) ).

% ent_disjI1_direct
tff(fact_822_ent__disjI2__direct,axiom,
    ! [B4: assn,A5: assn] : entails(B4,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B4)) ).

% ent_disjI2_direct
tff(fact_823_execute_Osimps,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_execute(A,heap_Time_Heap2(A,F3)) = F3 ).

% execute.simps
tff(fact_824_ent__false,axiom,
    ! [P: assn] : entails(bot_bot(assn),P) ).

% ent_false
tff(fact_825_ent__conjI,axiom,
    ! [A5: assn,B4: assn,C4: assn] :
      ( entails(A5,B4)
     => ( entails(A5,C4)
       => entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),B4),C4)) ) ) ).

% ent_conjI
tff(fact_826_ent__conjE1,axiom,
    ! [A5: assn,C4: assn,B4: assn] :
      ( entails(A5,C4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A5),B4),C4) ) ).

% ent_conjE1
tff(fact_827_ent__conjE2,axiom,
    ! [B4: assn,C4: assn,A5: assn] :
      ( entails(B4,C4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A5),B4),C4) ) ).

% ent_conjE2
tff(fact_828_ent__ex__preI,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] :
      ( ! [X3: A] : entails(aa(A,assn,P,X3),Q)
     => entails(ex_assn(A,P),Q) ) ).

% ent_ex_preI
tff(fact_829_ent__ex__postI,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),X: A] :
      ( entails(P,aa(A,assn,Q,X))
     => entails(P,ex_assn(A,Q)) ) ).

% ent_ex_postI
tff(fact_830_ent__wandI,axiom,
    ! [Q: assn,P: assn,R2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),P),R2)
     => entails(P,wand_assn(Q,R2)) ) ).

% ent_wandI
tff(fact_831_ent__mp,axiom,
    ! [P: assn,Q: assn] : entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),wand_assn(P,Q)),Q) ).

% ent_mp
tff(fact_832_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_833_execute__assert_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,H: heap_ext(product_unit)] :
      ( pp(aa(A,bool,P,X))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P,X)),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)),X),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),one_one(nat)))) ) ) ).

% execute_assert(1)
tff(fact_834_Heap__lub__empty,axiom,
    ! [A: $tType] : heap_Time_Heap_lub(A,bot_bot(set(heap_Time_Heap(A)))) = heap_Time_Heap2(A,aTP_Lamp_bu(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).

% Heap_lub_empty
tff(fact_835_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: bool] :
      ( entails(P,pure_assn(B2))
    <=> ( ! [H6: product_prod(heap_ext(product_unit),set(nat))] :
            ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H6))
           => pp(B2) )
        & entails(P,one_one(assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_836_pairself_Opelims,axiom,
    ! [B: $tType,A: $tType,X: fun(A,B),Xa2: product_prod(A,A),Y: product_prod(B,B)] :
      ( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X),Xa2) = Y )
     => ( pp(aa(product_prod(fun(A,B),product_prod(A,A)),bool,accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B)),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)),X),Xa2)))
       => ~ ! [A6: A,B5: A] :
              ( ( Xa2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5) )
             => ( ( Y = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X,A6)),aa(A,B,X,B5)) )
               => ~ pp(aa(product_prod(fun(A,B),product_prod(A,A)),bool,accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B)),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)),X),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)))) ) ) ) ) ).

% pairself.pelims
tff(fact_837_one__assn__raw_Osimps,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H),As)))
    <=> ( As = bot_bot(set(nat)) ) ) ).

% one_assn_raw.simps
tff(fact_838_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
      <=> pp(Y) )
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ( pp(Y)
            <=> ( As4 != bot_bot(set(nat)) ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_839_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ( As4 != bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_840_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
           => ( As4 = bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_841_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ).

% mult_le_cancel2
tff(fact_842_combine__options__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A)] : combine_options(A,F3,X,Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_bw(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F3),Y),X) ).

% combine_options_def
tff(fact_843_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N2),zero_zero(A)))
        <=> ( N2 = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_844_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N2))
        <=> ( N2 = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_845_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_846_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_847_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_848_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_849_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_850_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_851_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_852_less__nat__zero__code,axiom,
    ! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_853_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).

% neq0_conv
tff(fact_854_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A3)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_855_le0,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N2)) ).

% le0
tff(fact_856_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A3)) ).

% bot_nat_0.extremum
tff(fact_857_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N2 = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_858_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_859_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K) )
    <=> ( ( M = N2 )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_860_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2) )
    <=> ( ( M = N2 )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_861_mult__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_862_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N2 = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_863_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N2 = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_864_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) )
    <=> ( ( M = one_one(nat) )
        & ( N2 = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_865_combine__options__simps_I3_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),A3: A,B2: A] : combine_options(A,F3,aa(A,option(A),some(A),A3),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F3,A3),B2)) ).

% combine_options_simps(3)
tff(fact_866_combine__options__simps_I2_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A)] : combine_options(A,F3,X,none(A)) = X ).

% combine_options_simps(2)
tff(fact_867_combine__options__simps_I1_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Y: option(A)] : combine_options(A,F3,none(A),Y) = Y ).

% combine_options_simps(1)
tff(fact_868_pure__true,axiom,
    pure_assn(fTrue) = one_one(assn) ).

% pure_true
tff(fact_869_pure__assn__eq__emp__iff,axiom,
    ! [P: bool] :
      ( ( pure_assn(P) = one_one(assn) )
    <=> pp(P) ) ).

% pure_assn_eq_emp_iff
tff(fact_870_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_871_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_872_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_873_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_874_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_875_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_876_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_877_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_878_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_879_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_880_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_881_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_882_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_883_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_884_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_885_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_886_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_887_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% add_gr_0
tff(fact_888_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% mult_less_cancel2
tff(fact_889_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% nat_0_less_mult_iff
tff(fact_890_less__one,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),one_one(nat)))
    <=> ( N2 = zero_zero(nat) ) ) ).

% less_one
tff(fact_891_ent__pure__pre__iff__sng,axiom,
    ! [B2: bool,Q: assn] :
      ( entails(pure_assn(B2),Q)
    <=> ( pp(B2)
       => entails(one_one(assn),Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_892_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one_class.zero_le_one
tff(fact_893_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_894_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).

% not_one_le_zero
tff(fact_895_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one
tff(fact_896_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).

% not_one_less_zero
tff(fact_897_combine__options__assoc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A),Z2: option(A)] :
      ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Z3) = aa(A,A,aa(A,fun(A,A),F3,X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))
     => ( combine_options(A,F3,combine_options(A,F3,X,Y),Z2) = combine_options(A,F3,X,combine_options(A,F3,Y,Z2)) ) ) ).

% combine_options_assoc
tff(fact_898_combine__options__commute,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A)] :
      ( ! [X3: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F3,X3),Y3) = aa(A,A,aa(A,fun(A,A),F3,Y3),X3)
     => ( combine_options(A,F3,X,Y) = combine_options(A,F3,Y,X) ) ) ).

% combine_options_commute
tff(fact_899_combine__options__left__commute,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Y: option(A),X: option(A),Z2: option(A)] :
      ( ! [X3: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F3,X3),Y3) = aa(A,A,aa(A,fun(A,A),F3,Y3),X3)
     => ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Z3) = aa(A,A,aa(A,fun(A,A),F3,X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))
       => ( combine_options(A,F3,Y,combine_options(A,F3,X,Z2)) = combine_options(A,F3,X,combine_options(A,F3,Y,Z2)) ) ) ) ).

% combine_options_left_commute
tff(fact_900_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_901_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),X))
    <=> ( X != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_902_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) )
     => ( ( N2 = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_903_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F3: fun(fun(A,B),C),G3: C] :
          ( ! [X3: fun(A,B)] : aa(fun(A,B),C,F3,X3) = G3
         => ( aa(fun(A,B),C,F3,aTP_Lamp_bx(A,B)) = G3 ) ) ) ).

% fun_cong_unused_0
tff(fact_904_one__assn__def,axiom,
    one_one(assn) = abs_assn(one_assn_raw) ).

% one_assn_def
tff(fact_905_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3)) ) ) ) ).

% mult_left_le
tff(fact_906_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ) ).

% mult_le_one
tff(fact_907_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_908_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_909_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ).

% zero_less_two
tff(fact_910_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).

% zero_le
tff(fact_911_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_912_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_913_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N2))
        <=> ( N2 != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_914_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N2))
         => ( N2 != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_915_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N2),zero_zero(A))) ) ).

% not_less_zero
tff(fact_916_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N2: A] :
          ( ( N2 != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N2)) ) ) ).

% gr_zeroI
tff(fact_917_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) )
         => ( ( A3 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_918_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
         => ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_919_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_920_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_921_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_922_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,bool),X: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V,X3) = zero_zero(nat) )
         => pp(aa(A,bool,P,X3)) )
     => ( ! [X3: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X3)))
           => ( ~ pp(aa(A,bool,P,X3))
             => ? [Y4: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X3)))
                  & ~ pp(aa(A,bool,P,Y4)) ) ) )
       => pp(aa(A,bool,P,X)) ) ) ).

% infinite_descent0_measure
tff(fact_923_infinite__descent0,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
           => ( ~ pp(aa(nat,bool,P,N5))
             => ? [M4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N5))
                  & ~ pp(aa(nat,bool,P,M4)) ) ) )
       => pp(aa(nat,bool,P,N2)) ) ) ).

% infinite_descent0
tff(fact_924_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( N2 != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_925_less__zeroE,axiom,
    ! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),zero_zero(nat))) ).

% less_zeroE
tff(fact_926_not__less0,axiom,
    ! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),zero_zero(nat))) ).

% not_less0
tff(fact_927_not__gr0,axiom,
    ! [N2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
    <=> ( N2 = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_928_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).

% gr0I
tff(fact_929_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_930_nat__mult__1__right,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),one_one(nat)) = N2 ).

% nat_mult_1_right
tff(fact_931_nat__mult__1,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N2) = N2 ).

% nat_mult_1
tff(fact_932_le__0__eq,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),zero_zero(nat)))
    <=> ( N2 = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_933_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_934_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_935_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N2)) ).

% less_eq_nat.simps(1)
tff(fact_936_plus__nat_Oadd__0,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N2) = N2 ).

% plus_nat.add_0
tff(fact_937_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) = M )
     => ( N2 = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_938_mult__0,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N2) = zero_zero(nat) ).

% mult_0
tff(fact_939_assn__one__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),P) = P ).

% assn_one_left
tff(fact_940_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_941_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_942_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_943_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_944_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_945_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_946_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_947_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_948_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A3: A,Y: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_949_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_by(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_950_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_bz(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_951_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A3: A,Y: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_952_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).

% add_mono1
tff(fact_953_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))) ) ).

% less_add_one
tff(fact_954_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N2))) ) ) ) ).

% less_1_mult
tff(fact_955_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_956_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_957_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_nonpos
tff(fact_958_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_nonneg_nonneg
tff(fact_959_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_increasing2
tff(fact_960_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_961_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_increasing
tff(fact_962_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2)) ) ) ) ).

% add_decreasing
tff(fact_963_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_964_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_965_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_966_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_967_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_968_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_969_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ).

% split_mult_neg_le
tff(fact_970_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_le_0_iff
tff(fact_971_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono
tff(fact_972_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono_neg
tff(fact_973_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono
tff(fact_974_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_975_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono_neg
tff(fact_976_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ).

% split_mult_pos_le
tff(fact_977_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3))) ) ).

% zero_le_square
tff(fact_978_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_mono'
tff(fact_979_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_mono
tff(fact_980_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% pos_add_strict
tff(fact_981_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ ! [C3: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
               => ( C3 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_982_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_pos
tff(fact_983_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_984_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_985_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_986_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_987_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono
tff(fact_988_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_989_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_990_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono
tff(fact_991_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_992_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_993_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_994_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_995_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_996_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_997_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_998_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_999_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_1000_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_1001_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1002_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A))) ) ).

% not_square_less_zero
tff(fact_1003_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_1004_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R3: A,A3: A,B2: A,C2: A,D3: A] :
          ( ( R3 != zero_zero(A) )
         => ( ( ( A3 = B2 )
              & ( C2 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R3),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1005_ex__least__nat__le,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,N2))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N2))
            & ! [I: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K2))
               => ~ pp(aa(nat,bool,P,I)) )
            & pp(aa(nat,bool,P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_1006_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ? [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1007_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ) ).

% mult_less_mono1
tff(fact_1008_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ) ).

% mult_less_mono2
tff(fact_1009_pairself_Oelims,axiom,
    ! [B: $tType,A: $tType,X: fun(A,B),Xa2: product_prod(A,A),Y: product_prod(B,B)] :
      ( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X),Xa2) = Y )
     => ~ ! [A6: A,B5: A] :
            ( ( Xa2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5) )
           => ( Y != aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X,A6)),aa(A,B,X,B5)) ) ) ) ).

% pairself.elims
tff(fact_1010_pairself_Osimps,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: A,B2: A] : aa(product_prod(A,A),product_prod(B,B),pairself(A,B,F3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,A3)),aa(A,B,F3,B2)) ).

% pairself.simps
tff(fact_1011_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ca(nat,fun(nat,bool)),aTP_Lamp_cb(nat,fun(nat,bool)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_1012_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_strict_increasing2
tff(fact_1013_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).

% add_strict_increasing
tff(fact_1014_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_1015_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_1016_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_1017_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_1018_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1019_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1020_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1021_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1022_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1023_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1024_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1025_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1026_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1027_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1028_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1029_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1030_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1031_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1032_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).

% sum_squares_ge_zero
tff(fact_1033_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_1034_mod__emp__simp,axiom,
    ! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H),bot_bot(set(nat))))) ).

% mod_emp_simp
tff(fact_1035_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1036_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1037_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),one_one(A)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_mult_one_interval
tff(fact_1038_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ).

% nat_mult_le_cancel1
tff(fact_1039_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_1040_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1041_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1042_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1043_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1044_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X5)) ) ).

% linordered_field_no_lb
tff(fact_1045_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),X_1)) ) ).

% linordered_field_no_ub
tff(fact_1046_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mult_less_iff1
tff(fact_1047_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1048_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2) )
      <=> ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1049_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% nat_mult_less_cancel1
tff(fact_1050_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_epsilon
tff(fact_1051_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: A,R3: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R3))
        <=> ( R3 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1052_tap__def,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F3) = heap_Time_Heap2(A,aTP_Lamp_cc(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F3)) ).

% tap_def
tff(fact_1053_execute__tap,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),A),H: 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,F3)),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)),aa(heap_ext(product_unit),A,F3,H)),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),one_one(nat)))) ).

% execute_tap
tff(fact_1054_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
               => ( As4 = bot_bot(set(nat)) ) ) ) ) ) ).

% one_assn_raw.pelims(3)
tff(fact_1055_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))
               => ( As4 != bot_bot(set(nat)) ) ) ) ) ) ).

% one_assn_raw.pelims(2)
tff(fact_1056_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
      <=> pp(Y) )
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X))
       => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
             => ( ( pp(Y)
                <=> ( As4 = bot_bot(set(nat)) ) )
               => ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))) ) ) ) ) ).

% one_assn_raw.pelims(1)
tff(fact_1057_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% less_numeral_extra(1)
tff(fact_1058_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).

% less_numeral_extra(4)
tff(fact_1059_bounded__Max__nat,axiom,
    ! [P: fun(nat,bool),X: nat,M6: nat] :
      ( pp(aa(nat,bool,P,X))
     => ( ! [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M6)) )
       => ~ ! [M5: nat] :
              ( pp(aa(nat,bool,P,M5))
             => ~ ! [X5: nat] :
                    ( pp(aa(nat,bool,P,X5))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1060_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F2: fun(nat,fun(A,A)),A6: nat,B5: nat,Acc: A] : X != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A6),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B5),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1061_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),zero_zero(A))) ) ).

% le_numeral_extra(3)
tff(fact_1062_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),zero_zero(A))) ) ).

% less_numeral_extra(3)
tff(fact_1063_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).

% le_numeral_extra(4)
tff(fact_1064_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))))] :
      ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32)))))
     => ( ! [F2: fun(nat,fun(A,A)),A6: nat,B5: nat,Acc: A] :
            ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A6),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B5),Acc)))))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B5),A6))
               => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A6),one_one(nat))),B5),aa(A,A,aa(nat,fun(A,A),F2,A6),Acc))) )
             => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F2),A6),B5),Acc)) ) )
       => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,A0),A1),A22),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_1065_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_1066_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A3: nat,F3: fun(nat,fun(A,A)),Acc2: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = Acc2 ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A3),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_1067_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb,Xc) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
         => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_1068_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2))) ) ) ) ) ).

% power_strict_mono
tff(fact_1069_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ) ) ).

% power_increasing_iff
tff(fact_1070_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_1071_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_1072_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1073_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_1074_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_1075_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_1076_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ).

% times_divide_eq_left
tff(fact_1077_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
        | ( N2 = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1078_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1079_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1080_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1081_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1082_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1083_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),B2) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1084_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),A3) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1085_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2) )
          <=> ( M = N2 ) ) ) ) ).

% power_inject_exp
tff(fact_1086_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_1087_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_divide_1_iff
tff(fact_1088_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_1089_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_1090_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_1091_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_1092_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_1093_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_divide_1_iff
tff(fact_1094_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1095_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1096_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1097_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N2: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ) ).

% power_eq_0_iff
tff(fact_1098_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1099_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1100_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_1101_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N2: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) ) ) ) ).

% power_commuting_commutes
tff(fact_1102_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2)) ) ).

% power_mult_distrib
tff(fact_1103_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),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),N2)) ) ).

% power_commutes
tff(fact_1104_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_1105_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z2: A,W2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z2),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),W2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ).

% divide_divide_times_eq
tff(fact_1106_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z2: A,W2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z2),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W2)) ) ).

% times_divide_times_eq
tff(fact_1107_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2))) ) ) ) ).

% power_mono
tff(fact_1108_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ).

% zero_le_power
tff(fact_1109_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ).

% zero_less_power
tff(fact_1110_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ).

% one_le_power
tff(fact_1111_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1112_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_right_mono
tff(fact_1113_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1114_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1115_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1116_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1117_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1118_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2))) ) ) ) ).

% divide_right_mono_neg
tff(fact_1119_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_neg_neg
tff(fact_1120_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_1121_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_1122_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_pos_pos
tff(fact_1123_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_less_0_iff
tff(fact_1124_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_1125_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_1126_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_strict_right_mono
tff(fact_1127_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1128_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N2: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N2)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1129_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1130_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq
tff(fact_1131_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq
tff(fact_1132_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_1133_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 )
           => ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_1134_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1135_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1136_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: nat,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ).

% power_add
tff(fact_1137_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),N2)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% nat_power_less_imp_less
tff(fact_1138_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_1139_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_1140_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W2: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).

% frac_le
tff(fact_1141_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).

% frac_less
tff(fact_1142_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),Z2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).

% frac_less2
tff(fact_1143_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% divide_le_cancel
tff(fact_1144_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_1145_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonneg_pos
tff(fact_1146_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonpos_neg
tff(fact_1147_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_1148_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1149_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) ) ) ) ).

% div_positive
tff(fact_1150_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))) ) ) ).

% power_gt1_lemma
tff(fact_1151_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))) ) ) ).

% power_less_power_Suc
tff(fact_1152_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1153_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_1154_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_1155_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_1156_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% neg_less_divide_eq
tff(fact_1157_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% pos_divide_less_eq
tff(fact_1158_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_1159_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1160_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1161_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1162_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1163_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1164_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1165_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = B2 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1166_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = A3 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2)),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1167_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).

% add_frac_eq
tff(fact_1168_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1169_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1170_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y)),Z2) ) ) ) ).

% add_divide_eq_iff
tff(fact_1171_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% divide_add_eq_iff
tff(fact_1172_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1173_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N7))) ) ) ) ).

% power_strict_increasing
tff(fact_1174_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N7))) ) ) ) ).

% power_increasing
tff(fact_1175_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N2) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1176_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2)) ) ) ).

% gt_half_sum
tff(fact_1177_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) ) ) ).

% less_half_sum
tff(fact_1178_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) ) ) ).

% nat_mult_div_cancel1
tff(fact_1179_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb,Xc) = Y )
     => ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
               => ( Y = Xc ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
               => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) )
           => ~ pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_1180_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
      ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2)))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = Acc2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A3),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_1181_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ) ).

% power_Suc_less
tff(fact_1182_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1183_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1184_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1185_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_1186_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% pos_divide_le_eq
tff(fact_1187_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% neg_le_divide_eq
tff(fact_1188_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_1189_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_left_mono
tff(fact_1190_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_1191_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_1192_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1193_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1194_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N7)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1195_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N7)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ) ) ).

% power_decreasing
tff(fact_1196_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2) )
              <=> ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1197_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
               => ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1198_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1199_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ) ).

% self_le_power
tff(fact_1200_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ) ).

% one_less_power
tff(fact_1201_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self1
tff(fact_1202_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self2
tff(fact_1203_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self3
tff(fact_1204_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self4
tff(fact_1205_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),M)),N2) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1206_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),N2) = M ) ) ).

% div_mult_self_is_m
tff(fact_1207_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% div_mult_mult1
tff(fact_1208_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% div_mult_mult2
tff(fact_1209_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% div_mult_mult1_if
tff(fact_1210_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1211_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),K))) ) ).

% div_le_mono
tff(fact_1212_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)),M)) ).

% div_le_dividend
tff(fact_1213_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
        | ( N2 = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1214_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),N2)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)),I2)) ) ).

% less_mult_imp_div_less
tff(fact_1215_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2))),M)) ).

% times_div_less_eq_dividend
tff(fact_1216_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)),N2)),M)) ).

% div_times_less_eq_dividend
tff(fact_1217_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% div_greater_zero_iff
tff(fact_1218_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),N2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M))) ) ) ).

% div_le_mono2
tff(fact_1219_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q3)),N2))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3))) ) ) ).

% div_less_iff_less_mult
tff(fact_1220_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = M )
      <=> ( N2 = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1221_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)),M)) ) ) ).

% div_less_dividend
tff(fact_1222_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),Q3)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),N2)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1223_split__div,axiom,
    ! [P: fun(nat,bool),M: nat,N2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)))
    <=> ( ( ( N2 = zero_zero(nat) )
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ( ( N2 != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N2))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),I4)),J3) )
               => pp(aa(nat,bool,P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_1224_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)),N2)))) ) ).

% dividend_less_div_times
tff(fact_1225_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2))))) ) ).

% dividend_less_times_div
tff(fact_1226_verit__le__mono__div,axiom,
    ! [A5: nat,B4: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A5),B4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A5),N2)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B4,N2)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N2))) ) ) ).

% verit_le_mono_div
tff(fact_1227_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),X)),N2)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
            | ( N2 = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1228_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,C2: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),C2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_1229_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,A3: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_1230_split__div_H,axiom,
    ! [P: fun(nat,bool),M: nat,N2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)))
    <=> ( ( ( N2 = zero_zero(nat) )
          & pp(aa(nat,bool,P,zero_zero(nat))) )
        | ? [Q6: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q6)),M))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,suc,Q6))))
            & pp(aa(nat,bool,P,Q6)) ) ) ) ).

% split_div'
tff(fact_1231_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( 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),N2),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2) ) ) ) ).

% power_minus_mult
tff(fact_1232_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M: nat,P3: A] :
          ( ( ( M = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1233_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,N2: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_1234_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1235_minus__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A5: fun(A,B),B4: fun(A,B),X: 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)),A5),B4),X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A5,X)),aa(A,B,B4,X)) ) ).

% minus_apply
tff(fact_1236_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa(nat,nat,suc,Nat) = aa(nat,nat,suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_1237_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_1238_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N2) )
        <=> ( M = N2 ) ) ) ).

% of_nat_eq_iff
tff(fact_1239_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N2)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N2)) ) ) ).

% numeral_le_iff
tff(fact_1240_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N2)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2)) ) ) ).

% numeral_less_iff
tff(fact_1241_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W2: num,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2))),Z2) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1242_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N2)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ) ).

% numeral_times_numeral
tff(fact_1243_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2))) ) ).

% power_add_numeral
tff(fact_1244_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,N2: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N2))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2)))),B2) ) ).

% power_add_numeral2
tff(fact_1245_lessI,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,suc,N2))) ).

% lessI
tff(fact_1246_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N2))) ) ).

% Suc_mono
tff(fact_1247_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% Suc_less_eq
tff(fact_1248_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),aa(nat,nat,suc,M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M)) ) ).

% Suc_le_mono
tff(fact_1249_add__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) ).

% add_Suc_right
tff(fact_1250_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1251_diff__0__eq__0,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N2) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_1252_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N2)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),K) ).

% Suc_diff_diff
tff(fact_1253_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) ).

% diff_Suc_Suc
tff(fact_1254_diff__diff__cancel,axiom,
    ! [I2: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_1255_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1256_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( modulo_modulo(nat,M,N2) = M ) ) ).

% mod_less
tff(fact_1257_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1258_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1259_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1260_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1261_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_1262_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1263_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% distrib_left_numeral
tff(fact_1264_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B2: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).

% distrib_right_numeral
tff(fact_1265_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_1266_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B2: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).

% left_diff_distrib_numeral
tff(fact_1267_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_1268_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_1269_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = zero_zero(A) )
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_1270_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N2) )
        <=> ( zero_zero(nat) = N2 ) ) ) ).

% of_nat_0_eq_iff
tff(fact_1271_of__nat__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% of_nat_0
tff(fact_1272_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self1
tff(fact_1273_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self2
tff(fact_1274_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self3
tff(fact_1275_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self4
tff(fact_1276_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% of_nat_less_iff
tff(fact_1277_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ).

% of_nat_le_iff
tff(fact_1278_zero__less__Suc,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N2))) ).

% zero_less_Suc
tff(fact_1279_less__Suc0,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N2 = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_1280_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% of_nat_add
tff(fact_1281_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% zero_less_diff
tff(fact_1282_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_1283_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_1284_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% of_nat_mult
tff(fact_1285_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1286_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% diff_is_0_eq
tff(fact_1287_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N2) = one_one(A) )
        <=> ( N2 = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_1288_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N2) )
        <=> ( N2 = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_1289_of__nat__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),one_one(nat)) = one_one(A) ) ) ).

% of_nat_1
tff(fact_1290_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) ).

% mult_Suc_right
tff(fact_1291_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1292_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1293_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1294_diff__Suc__1,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N2)),one_one(nat)) = N2 ).

% diff_Suc_1
tff(fact_1295_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1296_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1297_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) = A3 )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1298_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W2: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) = B2 ) )
            & ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1299_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1300_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1301_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_1302_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_1303_Suc__pred,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(nat,nat,suc,zero_zero(nat)))) = N2 ) ) ).

% Suc_pred
tff(fact_1304_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N2)) ) ) ).

% one_le_mult_iff
tff(fact_1305_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1306_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1307_Suc__diff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_1308_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% of_nat_0_less_iff
tff(fact_1309_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) = N2 ) ) ).

% Suc_diff_1
tff(fact_1310_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),X)) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1311_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2))) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1312_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2))) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1313_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),X)) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1314_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I2: num,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),N2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),N2))) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1315_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,N2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),N2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),N2)),X)) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1316_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I2: num,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),N2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),N2))) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1317_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,N2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),N2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),N2)),X)) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1318_mod__induct,axiom,
    ! [P: fun(nat,bool),N2: nat,P3: nat,M: nat] :
      ( pp(aa(nat,bool,P,N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),P3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),P3))
         => ( ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),P3))
               => ( pp(aa(nat,bool,P,N5))
                 => pp(aa(nat,bool,P,modulo_modulo(nat,aa(nat,nat,suc,N5),P3))) ) )
           => pp(aa(nat,bool,P,M)) ) ) ) ) ).

% mod_induct
tff(fact_1319_mod__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
       => ( modulo_modulo(nat,M,N2) = M ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
       => ( modulo_modulo(nat,M,N2) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2),N2) ) ) ) ).

% mod_if
tff(fact_1320_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,N2))),N2)) ).

% mod_Suc_le_divisor
tff(fact_1321_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => ( modulo_modulo(nat,M,N2) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2),N2) ) ) ).

% le_mod_geq
tff(fact_1322_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N2))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) ) ) ).

% Suc_diff_Suc
tff(fact_1323_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),aa(nat,nat,suc,M))) ).

% diff_less_Suc
tff(fact_1324_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)) ) ) ).

% Suc_diff_le
tff(fact_1325_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A5: fun(A,B),B4: fun(A,B),X5: 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)),A5),B4),X5) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A5,X5)),aa(A,B,B4,X5)) ) ).

% fun_diff_def
tff(fact_1326_zero__induct__lemma,axiom,
    ! [P: fun(nat,bool),K: nat,I2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N5: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N5)))
           => pp(aa(nat,bool,P,N5)) )
       => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I2))) ) ) ).

% zero_induct_lemma
tff(fact_1327_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),K)),J) ).

% diff_commute
tff(fact_1328_n__not__Suc__n,axiom,
    ! [N2: nat] : N2 != aa(nat,nat,suc,N2) ).

% n_not_Suc_n
tff(fact_1329_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_1330_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N2: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)) ) ) ) ).

% of_nat_diff
tff(fact_1331_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( modulo_modulo(nat,M,N2) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2),N2) ) ) ).

% mod_geq
tff(fact_1332_Suc__to__right,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa(nat,nat,suc,N2) = M )
     => ( N2 = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_1333_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N2) ).

% diff_Suc_eq_diff_pred
tff(fact_1334_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_1335_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = modulo_modulo(A,A3,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1336_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1337_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1338_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = modulo_modulo(A,A3,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1339_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1340_diff__Suc__less,axiom,
    ! [N2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(nat,nat,suc,I2))),N2)) ) ).

% diff_Suc_less
tff(fact_1341_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_eq
tff(fact_1342_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A4: A,B2: A,B3: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A4,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B3,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1343_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_1344_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A3,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_1345_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_1346_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_1347_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).

% diff_mono
tff(fact_1348_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).

% diff_left_mono
tff(fact_1349_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% diff_right_mono
tff(fact_1350_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1351_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_1352_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D3),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).

% diff_strict_mono
tff(fact_1353_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3)) ) ) ) ).

% diff_eq_diff_less
tff(fact_1354_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).

% diff_strict_left_mono
tff(fact_1355_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% diff_strict_right_mono
tff(fact_1356_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib'
tff(fact_1357_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ).

% left_diff_distrib'
tff(fact_1358_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib
tff(fact_1359_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_1360_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
          ( ! [X3: A,K2: A] :
              ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
         => ( ! [X3: A,K2: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
           => ! [X5: A,K4: A] :
                ( ( pp(aa(A,bool,P,X5))
                  & pp(aa(A,bool,Q,X5)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))))
                  & pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1361_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
          ( ! [X3: A,K2: A] :
              ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
         => ( ! [X3: A,K2: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
           => ! [X5: A,K4: A] :
                ( ( pp(aa(A,bool,P,X5))
                  | pp(aa(A,bool,Q,X5)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))))
                  | pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1362_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N2)),M)) ).

% mod_less_eq_dividend
tff(fact_1363_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(2)
tff(fact_1364_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(3)
tff(fact_1365_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M) = zero_zero(nat) )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
tff(fact_1366_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1367_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ) ).

% diff_less_mono2
tff(fact_1368_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N2)),K)) ) ).

% less_imp_diff_less
tff(fact_1369_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero(nat) )
     => ? [M5: nat] : N2 = aa(nat,nat,suc,M5) ) ).

% not0_implies_Suc
tff(fact_1370_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_1371_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_1372_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_1373_zero__induct,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N5: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N5)))
           => pp(aa(nat,bool,P,N5)) )
       => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_1374_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N2: nat] :
      ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),zero_zero(nat)))
     => ( ! [Y3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y3)))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N2)) ) ) ) ).

% diff_induct
tff(fact_1375_nat__induct,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N5: nat] :
            ( pp(aa(nat,bool,P,N5))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N5))) )
       => pp(aa(nat,bool,P,N2)) ) ) ).

% nat_induct
tff(fact_1376_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1377_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_1378_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1379_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_1380_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_1381_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K))
     => ( ( K != aa(nat,nat,suc,I2) )
       => ~ ! [J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1382_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% Suc_lessD
tff(fact_1383_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K))
     => ~ ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1384_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( ( aa(nat,nat,suc,M) != N2 )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N2)) ) ) ).

% Suc_lessI
tff(fact_1385_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N2)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
       => ( M = N2 ) ) ) ).

% less_SucE
tff(fact_1386_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N2))) ) ).

% less_SucI
tff(fact_1387_Ex__less__Suc,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N2)))
          & pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,N2))
        | ? [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N2))
            & pp(aa(nat,bool,P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_1388_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
        | ( M = N2 ) ) ) ).

% less_Suc_eq
tff(fact_1389_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,suc,M))) ) ).

% not_less_eq
tff(fact_1390_All__less__Suc,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N2)))
         => pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,N2))
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N2))
           => pp(aa(nat,bool,P,I4)) ) ) ) ).

% All_less_Suc
tff(fact_1391_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),M))
    <=> ? [M7: nat] :
          ( ( M = aa(nat,nat,suc,M7) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M7)) ) ) ).

% Suc_less_eq2
tff(fact_1392_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,suc,M)))
       => ( M = N2 ) ) ) ).

% less_antisym
tff(fact_1393_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N2)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% Suc_less_SucD
tff(fact_1394_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K)) ) ) ).

% less_trans_Suc
tff(fact_1395_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( ! [I3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),aa(nat,nat,suc,I3)))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),J2))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J2),K2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I3),K2)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),J)) ) ) ) ).

% less_Suc_induct
tff(fact_1396_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( ! [I3: nat] :
            ( ( J = aa(nat,nat,suc,I3) )
           => pp(aa(nat,bool,P,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J))
             => ( pp(aa(nat,bool,P,aa(nat,nat,suc,I3)))
               => pp(aa(nat,bool,P,I3)) ) )
         => pp(aa(nat,bool,P,I2)) ) ) ) ).

% strict_inc_induct
tff(fact_1397_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,suc,M)))
      <=> ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1398_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K) )
        <=> ( M = N2 ) ) ) ) ).

% eq_diff_iff
tff(fact_1399_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ) ).

% le_diff_iff
tff(fact_1400_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1401_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),L))) ) ).

% diff_le_mono
tff(fact_1402_diff__le__self,axiom,
    ! [M: nat,N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),M)) ).

% diff_le_self
tff(fact_1403_le__diff__iff_H,axiom,
    ! [A3: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),C2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),C2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A3)) ) ) ) ).

% le_diff_iff'
tff(fact_1404_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ).

% diff_le_mono2
tff(fact_1405_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)),N2) = M ).

% diff_add_inverse2
tff(fact_1406_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M)),N2) = M ).

% diff_add_inverse
tff(fact_1407_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) ).

% diff_cancel2
tff(fact_1408_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2) ).

% Nat.diff_cancel
tff(fact_1409_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% Suc_leD
tff(fact_1410_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => ( M = aa(nat,nat,suc,N2) ) ) ) ).

% le_SucE
tff(fact_1411_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2))) ) ).

% le_SucI
tff(fact_1412_Suc__le__D,axiom,
    ! [N2: nat,M8: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),M8))
     => ? [M5: nat] : M8 = aa(nat,nat,suc,M5) ) ).

% Suc_le_D
tff(fact_1413_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
        | ( M = aa(nat,nat,suc,N2) ) ) ) ).

% le_Suc_eq
tff(fact_1414_Suc__n__not__le__n,axiom,
    ! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),N2)) ).

% Suc_n_not_le_n
tff(fact_1415_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),M)) ) ).

% not_less_eq_eq
tff(fact_1416_full__nat__induct,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M4)),N5))
             => pp(aa(nat,bool,P,M4)) )
         => pp(aa(nat,bool,P,N5)) )
     => pp(aa(nat,bool,P,N2)) ) ).

% full_nat_induct
tff(fact_1417_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( pp(aa(nat,bool,P,M))
       => ( ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N5))
             => ( pp(aa(nat,bool,P,N5))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N5))) ) )
         => pp(aa(nat,bool,P,N2)) ) ) ) ).

% nat_induct_at_least
tff(fact_1418_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R2: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),X3))
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),Y3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,Y3),Z3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),Z3)) ) )
         => ( ! [N5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,N5),aa(nat,nat,suc,N5)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,M),N2)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1419_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N2)) ).

% add_Suc_shift
tff(fact_1420_add__Suc,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) ).

% add_Suc
tff(fact_1421_nat__arith_Osuc1,axiom,
    ! [A5: nat,K: nat,A3: nat] :
      ( ( A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A3) )
     => ( aa(nat,nat,suc,A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A3)) ) ) ).

% nat_arith.suc1
tff(fact_1422_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K)) ).

% diff_mult_distrib
tff(fact_1423_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) ).

% diff_mult_distrib2
tff(fact_1424_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N2) )
    <=> ( M = N2 ) ) ).

% Suc_mult_cancel1
tff(fact_1425_left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,C2: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),C2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_1426_right__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,A3: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_1427_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,N2: nat] : modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_1428_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),aa(nat,nat,suc,zero_zero(nat)))),M)) ) ) ).

% nz_le_conv_less
tff(fact_1429_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( N2 = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1430_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1431_add__eq__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) = N2 ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N2)) ) ) ) ).

% add_eq_if
tff(fact_1432_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),M))
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1433_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),N2)) ) ) ) ).

% div_geq
tff(fact_1434_div__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
          | ( N2 = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
            | ( N2 = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),N2)) ) ) ) ).

% div_if
tff(fact_1435_nat__less__as__int,axiom,
    ! [X5: nat,Xa3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),Xa3))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ) ).

% nat_less_as_int
tff(fact_1436_nat__leq__as__int,axiom,
    ! [X5: nat,Xa3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Xa3))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ) ).

% nat_leq_as_int
tff(fact_1437_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),modulo_modulo(A,A3,B2)),A3)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1438_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1439_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1440_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% of_nat_0_le_iff
tff(fact_1441_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_1442_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ).

% mod_eqE
tff(fact_1443_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).

% le_iff_diff_le_0
tff(fact_1444_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),N2)) ) ) ) ).

% le_div_geq
tff(fact_1445_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).

% less_iff_diff_less_0
tff(fact_1446_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K: A,N2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),N2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),aa(A,A,aa(A,fun(A,A),minus_minus(A),N2),K))) ) ) ).

% add_le_imp_le_diff
tff(fact_1447_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K: A,N2: A,J: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),N2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),N2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N2),K)),J)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1448_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_le_eq
tff(fact_1449_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2)) ) ) ).

% le_diff_eq
tff(fact_1450_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),A3) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1451_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3))) ) ) ).

% le_add_diff
tff(fact_1452_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1453_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1454_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1455_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1456_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1457_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1458_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1459_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1460_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2))) ) ) ).

% less_imp_of_nat_less
tff(fact_1461_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ).

% of_nat_less_imp_less
tff(fact_1462_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I2: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I2)),aa(nat,A,semiring_1_of_nat(A),J))) ) ) ).

% of_nat_mono
tff(fact_1463_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% diff_shunt_var
tff(fact_1464_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1465_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_less_eq
tff(fact_1466_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2)) ) ) ).

% less_diff_eq
tff(fact_1467_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,N2: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% div_mult2_eq'
tff(fact_1468_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C2) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_1469_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_1470_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) ) ).

% square_diff_square_factored
tff(fact_1471_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),modulo_modulo(nat,M,N2)),N2)) ) ).

% mod_less_divisor
tff(fact_1472_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N2))) ) ).

% zero_le_numeral
tff(fact_1473_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N2)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_1474_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N2))) ) ).

% zero_less_numeral
tff(fact_1475_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N2)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_1476_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N2))) ) ).

% one_le_numeral
tff(fact_1477_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N2)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_1478_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N2: nat,N6: nat] :
          ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N2)),aa(nat,A,F3,N6))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_1479_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N2: nat,M: nat] :
          ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N2)),aa(nat,A,F3,M)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1480_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N2: nat,N6: nat] :
          ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,F3,N6))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_1481_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N2: nat,N6: nat] :
          ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N5))),aa(nat,A,F3,N5)))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N6)),aa(nat,A,F3,N2))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_1482_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),M)) ) ) ).

% diff_less
tff(fact_1483_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),A3) ) ).

% power_Suc2
tff(fact_1484_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ).

% power_Suc
tff(fact_1485_Ex__less__Suc2,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N2)))
          & pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        | ? [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N2))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1486_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
    <=> ? [M3: nat] : N2 = aa(nat,nat,suc,M3) ) ).

% gr0_conv_Suc
tff(fact_1487_All__less__Suc2,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N2)))
         => pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N2))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_1488_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ? [M5: nat] : N2 = aa(nat,nat,suc,M5) ) ).

% gr0_implies_Suc
tff(fact_1489_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N2)))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N2)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1490_diff__add__0,axiom,
    ! [N2: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1491_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ) ).

% less_diff_iff
tff(fact_1492_diff__less__mono,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),A3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2))) ) ) ).

% diff_less_mono
tff(fact_1493_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J)) ) ).

% less_diff_conv
tff(fact_1494_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1495_nat__compl__induct_H,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N5))
               => pp(aa(nat,bool,P,Nn)) )
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N5))) )
       => pp(aa(nat,bool,P,N2)) ) ) ).

% nat_compl_induct'
tff(fact_1496_nat__compl__induct,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N5))
               => pp(aa(nat,bool,P,Nn)) )
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N5))) )
       => pp(aa(nat,bool,P,N2)) ) ) ).

% nat_compl_induct
tff(fact_1497_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N2 = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_1498_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N2 = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_1499_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K))) ) ).

% le_diff_conv
tff(fact_1500_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1501_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1502_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1503_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1504_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N2)) ) ).

% Suc_leI
tff(fact_1505_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% Suc_le_eq
tff(fact_1506_dec__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,P,I2))
       => ( ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N5))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),J))
               => ( pp(aa(nat,bool,P,N5))
                 => pp(aa(nat,bool,P,aa(nat,nat,suc,N5))) ) ) )
         => pp(aa(nat,bool,P,J)) ) ) ) ).

% dec_induct
tff(fact_1507_inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,P,J))
       => ( ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N5))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),J))
               => ( pp(aa(nat,bool,P,aa(nat,nat,suc,N5)))
                 => pp(aa(nat,bool,P,N5)) ) ) )
         => pp(aa(nat,bool,P,I2)) ) ) ) ).

% inc_induct
tff(fact_1508_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% Suc_le_lessD
tff(fact_1509_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,suc,M)))
      <=> ( N2 = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1510_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% less_Suc_eq_le
tff(fact_1511_less__eq__Suc__le,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),M)) ) ).

% less_eq_Suc_le
tff(fact_1512_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N2))) ) ).

% le_imp_less_Suc
tff(fact_1513_nat__in__between__eq_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),aa(nat,nat,suc,A3))) )
    <=> ( B2 = A3 ) ) ).

% nat_in_between_eq(2)
tff(fact_1514_nat__in__between__eq_I1_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(nat,nat,suc,A3))) )
    <=> ( B2 = aa(nat,nat,suc,A3) ) ) ).

% nat_in_between_eq(1)
tff(fact_1515_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ~ ! [Q5: nat] : N2 != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)) ) ).

% less_natE
tff(fact_1516_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)))) ).

% less_add_Suc1
tff(fact_1517_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)))) ).

% less_add_Suc2
tff(fact_1518_less__iff__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
    <=> ? [K3: nat] : N2 = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1519_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ? [K2: nat] : N2 = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1520_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% Suc_mult_less_cancel1
tff(fact_1521_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1522_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% Suc_mult_le_cancel1
tff(fact_1523_mult__Suc,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) ).

% mult_Suc
tff(fact_1524_Suc__eq__plus1,axiom,
    ! [N2: nat] : aa(nat,nat,suc,N2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1525_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1526_Suc__eq__plus1__left,axiom,
    ! [N2: nat] : aa(nat,nat,suc,N2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N2) ).

% Suc_eq_plus1_left
tff(fact_1527_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),N2))) ).

% Suc_div_le_mono
tff(fact_1528_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( modulo_modulo(A,A3,B2) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1529_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A3,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1530_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_1531_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_1532_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2)) ) ).

% mod_div_decomp
tff(fact_1533_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2)) = A3 ) ).

% div_mult_mod_eq
tff(fact_1534_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = A3 ) ).

% mod_div_mult_eq
tff(fact_1535_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = A3 ) ).

% mod_mult_div_eq
tff(fact_1536_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),modulo_modulo(A,A3,B2)) = A3 ) ).

% mult_div_mod_eq
tff(fact_1537_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_mult1_eq
tff(fact_1538_le__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C2)),D3)) ) ) ).

% le_add_iff1
tff(fact_1539_le__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3))) ) ) ).

% le_add_iff2
tff(fact_1540_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C2)),D3)) ) ) ).

% less_add_iff1
tff(fact_1541_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3))) ) ) ).

% less_add_iff2
tff(fact_1542_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1543_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = A3 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2)),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1544_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1545_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y)),Z2) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1546_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1547_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N2)),N2)) ) ).

% mod_le_divisor
tff(fact_1548_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N2)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1549_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N2))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_1550_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1551_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1552_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N2,Q3) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
       => ~ ! [S5: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S5)) ) ) ).

% mod_eq_nat1E
tff(fact_1553_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N2,Q3) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => ~ ! [S5: nat] : N2 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S5)) ) ) ).

% mod_eq_nat2E
tff(fact_1554_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N2) = modulo_modulo(nat,Y,N2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
       => ? [Q5: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q5)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1555_div__less__mono,axiom,
    ! [A5: nat,B4: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A5),B4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => ( ( modulo_modulo(nat,A5,N2) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B4,N2) = zero_zero(nat) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A5),N2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N2))) ) ) ) ) ).

% div_less_mono
tff(fact_1556_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N2)))) ) ) ).

% power_gt1
tff(fact_1557_nat__diff__split,axiom,
    ! [P: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ! [D6: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
           => pp(aa(nat,bool,P,D6)) ) ) ) ).

% nat_diff_split
tff(fact_1558_nat__diff__split__asm,axiom,
    ! [P: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
            & ~ pp(aa(nat,bool,P,zero_zero(nat))) )
          | ? [D6: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
              & ~ pp(aa(nat,bool,P,D6)) ) ) ) ).

% nat_diff_split_asm
tff(fact_1559_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K))) ) ) ).

% less_diff_conv2
tff(fact_1560_ex__least__nat__less,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,N2))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),N2))
            & ! [I: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K2))
               => ~ pp(aa(nat,bool,P,I)) )
            & pp(aa(nat,bool,P,aa(nat,nat,suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_1561_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),M))) ) ) ).

% n_less_n_mult_m
tff(fact_1562_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2))) ) ) ).

% n_less_m_mult_n
tff(fact_1563_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2))) ) ) ).

% one_less_mult
tff(fact_1564_nat__induct__non__zero,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
             => ( pp(aa(nat,bool,P,N5))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N5))) ) )
         => pp(aa(nat,bool,P,N2)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1565_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),M) = N2 ) ) ) ).

% nat_eq_add_iff1
tff(fact_1566_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),N2) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1567_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),M)),N2)) ) ) ).

% nat_le_add_iff1
tff(fact_1568_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),N2))) ) ) ).

% nat_le_add_iff2
tff(fact_1569_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),M)),N2) ) ) ).

% nat_diff_add_eq1
tff(fact_1570_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),N2)) ) ) ).

% nat_diff_add_eq2
tff(fact_1571_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N2),K))) ) ).

% power_gt_expt
tff(fact_1572_nat__one__le__power,axiom,
    ! [I2: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),N2))) ) ).

% nat_one_le_power
tff(fact_1573_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_1574_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_1575_split__mod,axiom,
    ! [P: fun(nat,bool),M: nat,N2: nat] :
      ( pp(aa(nat,bool,P,modulo_modulo(nat,M,N2)))
    <=> ( ( ( N2 = zero_zero(nat) )
         => pp(aa(nat,bool,P,M)) )
        & ( ( N2 != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N2))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),I4)),J3) )
               => pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_1576_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1577_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1578_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N2))),A3)) ) ) ) ).

% power_Suc_le_self
tff(fact_1579_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N2))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_1580_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,N2: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ) ) ) ).

% power_diff
tff(fact_1581_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),M)),N2)) ) ) ).

% nat_less_add_iff1
tff(fact_1582_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),N2))) ) ) ).

% nat_less_add_iff2
tff(fact_1583_mult__eq__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = zero_zero(nat) ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N2)) ) ) ) ).

% mult_eq_if
tff(fact_1584_div__nat__eqI,axiom,
    ! [N2: nat,Q3: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,suc,Q3))))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_1585_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),C2))),modulo_modulo(A,A3,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1586_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R3: A,S2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R3),S2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(A,A,aa(A,fun(A,A),minus_minus(A),V2),U))),S2))),V2)) ) ) ) ) ).

% scaling_mono
tff(fact_1587_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1588_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
         => ~ ! [N5: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N5)))),E3)) ) ) ).

% nat_approx_posE
tff(fact_1589_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N5)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_1590_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N2: nat] : aa(nat,A,semiring_1_of_nat(A),N2) = semiri8178284476397505188at_aux(A,aTP_Lamp_cd(A,A),N2,zero_zero(A)) ) ).

% of_nat_code
tff(fact_1591_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N2: nat] :
      ( ! [M5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),zero_zero(nat)))
     => ( ! [M5: nat,N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N5),modulo_modulo(nat,M5,N5)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),N5)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N2)) ) ) ).

% gcd_nat_induct
tff(fact_1592_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X2: A] : aa(option(A),nat,size_option(A,X),aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1593_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N2: nat,A3: A] :
          ( ( ( N2 = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N2) = one_one(A) ) )
          & ( ( N2 != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N2) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ce(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_1594_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(option(A),nat,size_option(A,X),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_1595_Diff__cancel,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),A5) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1596_empty__Diff,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A5) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1597_Diff__empty,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),bot_bot(set(A))) = A5 ).

% Diff_empty
tff(fact_1598_Un__Diff__cancel2,axiom,
    ! [A: $tType,B4: set(A),A5: 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)),B4),A5)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),A5) ).

% Un_Diff_cancel2
tff(fact_1599_Un__Diff__cancel,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) ).

% Un_Diff_cancel
tff(fact_1600_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ).

% Diff_eq_empty_iff
tff(fact_1601_Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_1602_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1603_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1604_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1605_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1606_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1607_verit__le__mono__div__int,axiom,
    ! [A5: int,B4: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A5),B4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N2)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B4,N2)),zero_zero(int)),one_one(int),zero_zero(int)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N2))) ) ) ).

% verit_le_mono_div_int
tff(fact_1608_zdiv__mono__strict,axiom,
    ! [A5: int,B4: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A5),B4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N2))
       => ( ( modulo_modulo(int,A5,N2) = zero_zero(int) )
         => ( ( modulo_modulo(int,B4,N2) = zero_zero(int) )
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N2))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1609_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N2),K)),modulo_modulo(int,N2,K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
              & ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I4),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_1610_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N2),K)),modulo_modulo(int,N2,K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
              & ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I4),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_1611_zmod__zmult2__eq,axiom,
    ! [C2: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C2))
     => ( modulo_modulo(int,A3,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2),C2))),modulo_modulo(int,A3,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1612_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1613_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int))) ) ).

% neg_mod_sign
tff(fact_1614_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,K,L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1615_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),modulo_modulo(int,K,L))) ) ).

% neg_mod_bound
tff(fact_1616_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1617_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,M,K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_1618_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1619_neg__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,A3,B2)),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),modulo_modulo(int,A3,B2))) ) ) ).

% neg_mod_conj
tff(fact_1620_pos__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A3,B2)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,A3,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_1621_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( modulo_modulo(int,I2,K) = I2 )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2)) ) ) ) ).

% zmod_trivial_iff
tff(fact_1622_plusinfinity,axiom,
    ! [D3: int,P6: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P6,X3))
          <=> pp(aa(int,bool,P6,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),X3))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P6,X3)) ) )
         => ( ? [X_13: int] : pp(aa(int,bool,P6,X_13))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_1623_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P1,X3))
          <=> pp(aa(int,bool,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z5))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P1,X3)) ) )
         => ( ? [X_13: int] : pp(aa(int,bool,P1,X_13))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_1624_decr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D3))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X5: int] :
              ( pp(aa(int,bool,P,X5))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1625_int__mod__pos__eq,axiom,
    ! [A3: int,B2: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( modulo_modulo(int,A3,B2) = R3 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1626_int__mod__neg__eq,axiom,
    ! [A3: int,B2: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( modulo_modulo(int,A3,B2) = R3 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1627_split__zmod,axiom,
    ! [P: fun(int,bool),N2: int,K: int] :
      ( pp(aa(int,bool,P,modulo_modulo(int,N2,K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,N2)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_1628_q__pos__lemma,axiom,
    ! [B3: int,Q7: int,R6: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q7)),R6)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R6),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q7)) ) ) ) ).

% q_pos_lemma
tff(fact_1629_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B3: int,Q7: int,R6: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q7)),R6) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q7)),R6)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R6),B3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q3),Q7)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1630_incr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D3))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X5: int] :
              ( pp(aa(int,bool,P,X5))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1631_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B3: int,Q7: int,R6: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q7)),R6) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q7)),R6)),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R6))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q7),Q3)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1632_unique__quotient__lemma,axiom,
    ! [B2: int,Q7: int,R6: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q7)),R6)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R6),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q7),Q3)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1633_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q7: int,R6: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q7)),R6)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R6))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q3),Q7)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1634_imp__le__cong,axiom,
    ! [X: int,X8: int,P: bool,P6: bool] :
      ( ( X = X8 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
         => ( pp(P)
          <=> pp(P6) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
           => pp(P) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
           => pp(P6) ) ) ) ) ).

% imp_le_cong
tff(fact_1635_conj__le__cong,axiom,
    ! [X: int,X8: int,P: bool,P6: bool] :
      ( ( X = X8 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
         => ( pp(P)
          <=> pp(P6) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
            & pp(P) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
            & pp(P6) ) ) ) ) ).

% conj_le_cong
tff(fact_1636_verit__la__generic,axiom,
    ! [A3: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),X))
      | ( A3 = X )
      | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),A3)) ) ).

% verit_la_generic
tff(fact_1637_split__zdiv,axiom,
    ! [P: fun(int,bool),N2: int,K: int] :
      ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N2),K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,zero_zero(int))) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,I4)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_1638_int__div__neg__eq,axiom,
    ! [A3: int,B2: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1639_int__div__pos__eq,axiom,
    ! [A3: int,B2: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1640_zdiv__zmult2__eq,axiom,
    ! [C2: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1641_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1642_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1643_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1644_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I2),K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1645_div__nonpos__pos__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).

% div_nonpos_pos_le0
tff(fact_1646_div__nonneg__neg__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).

% div_nonneg_neg_le0
tff(fact_1647_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))) ) ) ).

% div_positive_int
tff(fact_1648_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ) ).

% div_int_pos_iff
tff(fact_1649_zdiv__mono2__neg,axiom,
    ! [A3: int,B3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1650_zdiv__mono1__neg,axiom,
    ! [A3: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_1651_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I2),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1652_zdiv__mono2,axiom,
    ! [A3: int,B3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))) ) ) ) ).

% zdiv_mono2
tff(fact_1653_zdiv__mono1,axiom,
    ! [A3: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2))) ) ) ).

% zdiv_mono1
tff(fact_1654_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1655_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1656_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_1657_div__neg__pos__less0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).

% div_neg_pos_less0
tff(fact_1658_Diff__mono,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),D4: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D4),B4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),D4))) ) ) ).

% Diff_mono
tff(fact_1659_Diff__subset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)),A5)) ).

% Diff_subset
tff(fact_1660_double__diff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),A5)) = A5 ) ) ) ).

% double_diff
tff(fact_1661_int__ops_I6_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = zero_zero(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ) ) ).

% int_ops(6)
tff(fact_1662_Diff__Int__distrib2,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),C4) = 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)),A5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) ).

% Diff_Int_distrib2
tff(fact_1663_Diff__Int__distrib,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = 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)),C4),A5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),B4)) ).

% Diff_Int_distrib
tff(fact_1664_Diff__Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ).

% Diff_Diff_Int
tff(fact_1665_Diff__Int2,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B4: 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)),A5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) = 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)),A5),C4)),B4) ).

% Diff_Int2
tff(fact_1666_Int__Diff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C4)) ).

% Int_Diff
tff(fact_1667_set__diff__diff__left,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4)) ).

% set_diff_diff_left
tff(fact_1668_Un__Diff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: 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)),A5),B4)),C4) = 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)),A5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C4)) ).

% Un_Diff
tff(fact_1669_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
     => ? [B5: A] : pp(aa(set(A),bool,member(A,B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5))) ) ).

% psubset_imp_ex_mem
tff(fact_1670_subset__minus__empty,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_1671_disjoint__alt__simp2,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) != A5 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp2
tff(fact_1672_disjoint__alt__simp1,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) = A5 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) ) ) ).

% disjoint_alt_simp1
tff(fact_1673_Int__Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B4: 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_1674_Diff__triv,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) = A5 ) ) ).

% Diff_triv
tff(fact_1675_Diff__partition,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)) = B4 ) ) ).

% Diff_partition
tff(fact_1676_Diff__subset__conv,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)),C4))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4))) ) ).

% Diff_subset_conv
tff(fact_1677_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N2))) ) ) ).

% pochhammer_pos
tff(fact_1678_Un__Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B4: 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) = A5 ).

% Un_Diff_Int
tff(fact_1679_Int__Diff__Un,axiom,
    ! [A: $tType,A5: set(A),B4: 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = A5 ).

% Int_Diff_Un
tff(fact_1680_Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) = 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),C4)) ).

% Diff_Int
tff(fact_1681_Diff__Un,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4)) = 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)),A5),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),C4)) ).

% Diff_Un
tff(fact_1682_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat,N2: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( comm_s3205402744901411588hammer(A,A3,N2) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1683_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N2: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N2) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( comm_s3205402744901411588hammer(A,A3,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1684_zdiff__int__split,axiom,
    ! [P: fun(int,bool),X: nat,Y: nat] :
      ( pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Y))))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y)))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
         => pp(aa(int,bool,P,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_1685_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I2: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I2) = I2 ) ).

% of_nat_aux.simps(1)
tff(fact_1686_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N2: nat,I2: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N2),I2) = semiri8178284476397505188at_aux(A,Inc,N2,aa(A,A,Inc,I2)) ) ).

% of_nat_aux.simps(2)
tff(fact_1687_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N2))) ) ) ).

% pochhammer_nonneg
tff(fact_1688_disjoint__alt__simp3,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)),A5))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp3
tff(fact_1689_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N2: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N2)) = 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)),N2)) ) ).

% pochhammer_rec
tff(fact_1690_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,N2: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N2)) = 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),N2))),comm_s3205402744901411588hammer(A,Z2,N2)) ) ).

% pochhammer_rec'
tff(fact_1691_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N2: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% pochhammer_Suc
tff(fact_1692_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,N2: nat,M: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,N2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N2)),M)) ) ).

% pochhammer_product'
tff(fact_1693_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N2: nat,Z2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( comm_s3205402744901411588hammer(A,Z2,N2) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ) ).

% pochhammer_product
tff(fact_1694_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N5))) ) ).

% real_arch_simple
tff(fact_1695_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N5))) ) ).

% reals_Archimedean2
tff(fact_1696_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ).

% zle_diff1_eq
tff(fact_1697_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2)) ) ).

% zle_add1_eq_le
tff(fact_1698_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
          & ( K = aa(nat,int,semiring_1_of_nat(int),N5) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1699_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ~ ! [N5: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N5) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5)) ) ) ).

% pos_int_cases
tff(fact_1700_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1701_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ~ ! [N5: nat] : K != aa(nat,int,semiring_1_of_nat(int),N5) ) ).

% nonneg_int_cases
tff(fact_1702_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ? [N5: nat] : K = aa(nat,int,semiring_1_of_nat(int),N5) ) ).

% zero_le_imp_eq_int
tff(fact_1703_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% zle_int
tff(fact_1704_Diff__idemp,axiom,
    ! [A: $tType,A5: set(A),B4: 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)),A5),B4)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) ).

% Diff_idemp
tff(fact_1705_Diff__iff,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)))
    <=> ( pp(aa(set(A),bool,member(A,C2),A5))
        & ~ pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% Diff_iff
tff(fact_1706_DiffI,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),A5))
     => ( ~ pp(aa(set(A),bool,member(A,C2),B4))
       => pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4))) ) ) ).

% DiffI
tff(fact_1707_minus__set__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B4))) ).

% minus_set_def
tff(fact_1708_DiffD2,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)))
     => ~ pp(aa(set(A),bool,member(A,C2),B4)) ) ).

% DiffD2
tff(fact_1709_DiffD1,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)))
     => pp(aa(set(A),bool,member(A,C2),A5)) ) ).

% DiffD1
tff(fact_1710_DiffE,axiom,
    ! [A: $tType,C2: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)))
     => ~ ( pp(aa(set(A),bool,member(A,C2),A5))
         => pp(aa(set(A),bool,member(A,C2),B4)) ) ) ).

% DiffE
tff(fact_1711_set__diff__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_cf(set(A),fun(set(A),fun(A,bool)),A5),B4)) ).

% set_diff_eq
tff(fact_1712_int__ge__induct,axiom,
    ! [K: int,I2: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I3))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_ge_induct
tff(fact_1713_int__le__induct,axiom,
    ! [I2: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),K))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_le_induct
tff(fact_1714_int__induct,axiom,
    ! [P: fun(int,bool),K: int,I2: int] :
      ( pp(aa(int,bool,P,K))
     => ( ! [I3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I3))
           => ( pp(aa(int,bool,P,I3))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_induct
tff(fact_1715_int__gr__induct,axiom,
    ! [K: int,I2: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I3))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_gr_induct
tff(fact_1716_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2))
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
tff(fact_1717_int__less__induct,axiom,
    ! [I2: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I3),K))
             => ( pp(aa(int,bool,P,I3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_less_induct
tff(fact_1718_less__eq__int__code_I1_J,axiom,
    pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).

% less_eq_int_code(1)
tff(fact_1719_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)),Z2)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int))) ) ).

% odd_less_0_iff
tff(fact_1720_less__int__code_I1_J,axiom,
    ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),zero_zero(int))) ).

% less_int_code(1)
tff(fact_1721_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_1722_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N2) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N2 = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1723_add1__zle__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ).

% add1_zle_eq
tff(fact_1724_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z2)) ) ).

% zless_imp_add1_zle
tff(fact_1725_zle__iff__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2))
    <=> ? [N: nat] : Z2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% zle_iff_zadd
tff(fact_1726_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2))) ) ).

% le_imp_0_less
tff(fact_1727_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ).

% int_one_le_iff_zero_less
tff(fact_1728_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2))
    <=> ? [N: nat] : Z2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% zless_iff_Suc_zadd
tff(fact_1729_nat0__intermed__int__val,axiom,
    ! [N2: nat,F3: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F3,N2)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N2))
              & ( aa(nat,int,F3,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1730_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R3) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),L)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q3 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1731_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).

% pochhammer_minus'
tff(fact_1732_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),X)) = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_1733_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,uminus_uminus(A),X) = aa(A,A,uminus_uminus(A),Y) )
        <=> ( X = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_1734_uminus__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A5: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A5),X) = aa(B,B,uminus_uminus(B),aa(A,B,A5,X)) ) ).

% uminus_apply
tff(fact_1735_Compl__anti__mono,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B4)),aa(set(A),set(A),uminus_uminus(set(A)),A5))) ) ).

% Compl_anti_mono
tff(fact_1736_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B4)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5)) ) ).

% Compl_subset_Compl_iff
tff(fact_1737_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% neg_le_iff_le
tff(fact_1738_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% compl_le_compl_iff
tff(fact_1739_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% neg_less_iff_less
tff(fact_1740_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% compl_less_compl_iff
tff(fact_1741_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% mult_minus_left
tff(fact_1742_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) ) ).

% minus_mult_minus
tff(fact_1743_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% mult_minus_right
tff(fact_1744_Compl__disjoint2,axiom,
    ! [A: $tType,A5: 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)),A5)),A5) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_1745_Compl__disjoint,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_1746_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% abs_mult_self_eq
tff(fact_1747_inter__compl__diff__conv,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) ).

% inter_compl_diff_conv
tff(fact_1748_Diff__Compl,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ).

% Diff_Compl
tff(fact_1749_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N2)) = aa(nat,A,semiring_1_of_nat(A),N2) ) ).

% abs_of_nat
tff(fact_1750_Compl__Diff__eq,axiom,
    ! [A: $tType,A5: set(A),B4: 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)),A5),B4)) = 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)),A5)),B4) ).

% Compl_Diff_eq
tff(fact_1751_negative__zle,axiom,
    ! [N2: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zle
tff(fact_1752_mod__not__dist,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,uminus_uminus(assn),P)),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H))
        & ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H)) ) ) ).

% mod_not_dist
tff(fact_1753_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_less_eq_nonneg
tff(fact_1754_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_1755_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_le_0_iff_le
tff(fact_1756_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_1757_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_0_iff_less
tff(fact_1758_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_1759_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_pos
tff(fact_1760_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_1761_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_1762_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_1763_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_1764_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% abs_le_self_iff
tff(fact_1765_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_1766_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_1767_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),X)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_1768_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),X) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_1769_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),X))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_1770_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_1771_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_1772_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_1773_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_1774_negative__zless,axiom,
    ! [N2: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2)))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zless
tff(fact_1775_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_1776_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_1777_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B2))),zero_zero(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_1778_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N2: 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))),N2)),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))),N2)),A3)) = A3 ) ).

% left_minus_one_mult_self
tff(fact_1779_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N2: 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))),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_1780_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2))),Y) ) ).

% semiring_norm(172)
tff(fact_1781_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2)))),Y) ) ).

% semiring_norm(171)
tff(fact_1782_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2)))),Y) ) ).

% semiring_norm(170)
tff(fact_1783_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1784_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1785_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1786_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),Z2)),one_one(int)))
    <=> ( Z2 = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_1787_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N2),M)) ) ) ).

% neg_numeral_le_iff
tff(fact_1788_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N2),M)) ) ) ).

% neg_numeral_less_iff
tff(fact_1789_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_1790_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_1791_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W2: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B2 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1792_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A3 )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1793_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_1794_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_1795_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N2)))
        <=> ( ( A3 != zero_zero(A) )
            | ( N2 = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1796_small__lazy_H_Ocases,axiom,
    ! [X: product_prod(int,int)] :
      ~ ! [D2: int,I3: int] : X != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I3) ).

% small_lazy'.cases
tff(fact_1797_exhaustive__int_H_Ocases,axiom,
    ! [X: product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int))] :
      ~ ! [F2: fun(int,option(product_prod(bool,list(code_term)))),D2: int,I3: int] : X != aa(product_prod(int,int),product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int)),aa(fun(int,option(product_prod(bool,list(code_term)))),fun(product_prod(int,int),product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int))),product_Pair(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int)),F2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I3)) ).

% exhaustive_int'.cases
tff(fact_1798_full__exhaustive__int_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int))] :
      ~ ! [F2: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: int,I3: int] : X != aa(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int)),aa(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),fun(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int))),product_Pair(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int)),F2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I3)) ).

% full_exhaustive_int'.cases
tff(fact_1799_unique__remainder,axiom,
    ! [A3: int,B2: int,Q3: int,R3: int,Q7: int,R6: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),R6))
       => ( R3 = R6 ) ) ) ).

% unique_remainder
tff(fact_1800_unique__quotient,axiom,
    ! [A3: int,B2: int,Q3: int,R3: int,Q7: int,R6: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),R6))
       => ( Q3 = Q7 ) ) ) ).

% unique_quotient
tff(fact_1801_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A5: fun(A,B),X5: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A5),X5) = aa(B,B,uminus_uminus(B),aa(A,B,A5,X5)) ) ).

% fun_Compl_def
tff(fact_1802_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_minus_self
tff(fact_1803_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_1804_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ).

% abs_le_D2
tff(fact_1805_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)) ) ) ) ).

% abs_leI
tff(fact_1806_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_1807_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
            & ( ( A3 = B2 )
              | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1808_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & ( ( B2 = A3 )
              | ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1809_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A))) ) ).

% abs_minus_le_zero
tff(fact_1810_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ) ).

% abs_if
tff(fact_1811_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X5: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X5) = aa(A,A,uminus_uminus(A),X5) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X5) = X5 ) ) ) ) ).

% abs_if_raw
tff(fact_1812_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_1813_zabs__def,axiom,
    ! [I2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I2) = aa(int,int,uminus_uminus(int),I2) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I2) = I2 ) ) ) ).

% zabs_def
tff(fact_1814_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% abs_le_D1
tff(fact_1815_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_self
tff(fact_1816_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_1817_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_imp_neg_le
tff(fact_1818_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_le_iff
tff(fact_1819_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_minus_iff
tff(fact_1820_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_le_swap2
tff(fact_1821_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_le_swap1
tff(fact_1822_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_1823_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_less_iff
tff(fact_1824_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% less_minus_iff
tff(fact_1825_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% verit_negate_coefficient(2)
tff(fact_1826_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_less_swap1
tff(fact_1827_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_less_swap2
tff(fact_1828_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A3 = B2 )
            | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_1829_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_1830_minus__assn__def,axiom,
    ! [A3: assn,B2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),minus_minus(assn),A3),B2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),aa(assn,assn,uminus_uminus(assn),B2)) ).

% minus_assn_def
tff(fact_1831_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_cg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,bool),set(A),collect(A),P))),aa(fun(A,bool),set(A),collect(A),Q)) ).

% Collect_imp_eq
tff(fact_1832_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_1833_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q3 ) ) ).

% div_int_unique
tff(fact_1834_zminus1__lemma,axiom,
    ! [A3: int,B2: int,Q3: int,R3: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R3),zero_zero(int)),aa(int,int,uminus_uminus(int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R3),zero_zero(int)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R3)))) ) ) ).

% zminus1_lemma
tff(fact_1835_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
     => ( modulo_modulo(int,K,L) = R3 ) ) ).

% mod_int_unique
tff(fact_1836_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_zero
tff(fact_1837_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_1838_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_1839_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq
tff(fact_1840_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).

% abs_triangle_ineq2
tff(fact_1841_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).

% abs_triangle_ineq3
tff(fact_1842_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))) ) ).

% abs_triangle_ineq2_sym
tff(fact_1843_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3))) ) ) ) ).

% abs_mult_less
tff(fact_1844_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N2))) ) ).

% neg_numeral_le_numeral
tff(fact_1845_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_1846_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% le_minus_one_simps(2)
tff(fact_1847_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(4)
tff(fact_1848_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N2))) ) ).

% neg_numeral_less_numeral
tff(fact_1849_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_1850_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% less_minus_one_simps(2)
tff(fact_1851_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(4)
tff(fact_1852_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W2: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).

% numeral_times_minus_swap
tff(fact_1853_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
        <=> ( ( X = one_one(A) )
            | ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1854_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y)) ) ).

% diff_eq
tff(fact_1855_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_1856_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_1857_subset__Compl__self__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_1858_Compl__Int,axiom,
    ! [A: $tType,A5: set(A),B4: 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)),A5),B4)) = 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)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B4)) ).

% Compl_Int
tff(fact_1859_Compl__Un,axiom,
    ! [A: $tType,A5: set(A),B4: 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)),A5),B4)) = 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)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B4)) ).

% Compl_Un
tff(fact_1860_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N2)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% not_int_zless_negative
tff(fact_1861_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E2)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_1862_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_1863_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_1864_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))))) ) ).

% abs_diff_triangle_ineq
tff(fact_1865_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq4
tff(fact_1866_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R3))) ) ) ) ).

% abs_diff_le_iff
tff(fact_1867_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R3))) ) ) ) ).

% abs_diff_less_iff
tff(fact_1868_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% abs_div_pos
tff(fact_1869_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N2))) ) ).

% zero_le_power_abs
tff(fact_1870_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2)))) ) ).

% not_zero_le_neg_numeral
tff(fact_1871_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_1872_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% le_minus_one_simps(1)
tff(fact_1873_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(3)
tff(fact_1874_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2)))) ) ).

% not_zero_less_neg_numeral
tff(fact_1875_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_1876_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% less_minus_one_simps(1)
tff(fact_1877_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(3)
tff(fact_1878_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_le_neg_numeral
tff(fact_1879_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_1880_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_1881_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_le_numeral
tff(fact_1882_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_1883_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_1884_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_less_neg_numeral
tff(fact_1885_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_1886_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_less_numeral
tff(fact_1887_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_1888_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1889_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1890_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A3 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_1891_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_1892_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% inf_shunt
tff(fact_1893_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q3: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q3))),R3)) ) ) ).

% sup_neg_inf
tff(fact_1894_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2))) ) ) ).

% shunt2
tff(fact_1895_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z2))) ) ) ).

% shunt1
tff(fact_1896_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N2) = 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))),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ).

% power_minus
tff(fact_1897_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1898_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B4))) ) ).

% disjoint_eq_subset_Compl
tff(fact_1899_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L))) ) ).

% abs_mod_less
tff(fact_1900_int__cases4,axiom,
    ! [M: int] :
      ( ! [N5: nat] : M != aa(nat,int,semiring_1_of_nat(int),N5)
     => ~ ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N5)) ) ) ) ).

% int_cases4
tff(fact_1901_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N2)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))))
    <=> ( ( N2 = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_1902_negative__zle__0,axiom,
    ! [N2: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2))),zero_zero(int))) ).

% negative_zle_0
tff(fact_1903_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ~ ! [N5: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N5)) ) ).

% nonpos_int_cases
tff(fact_1904_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)),modulo_modulo(int,K,L))) ).

% eucl_rel_int
tff(fact_1905_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X)))) ) ).

% abs_add_one_gt_zero
tff(fact_1906_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1907_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1908_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1909_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1910_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_1911_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_1912_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_1913_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_1914_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = B2 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1915_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1916_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,X: A,Y: A] :
          ( ( Z2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1917_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_1918_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_1919_uminus__assn__def,axiom,
    ! [P: assn] : aa(assn,assn,uminus_uminus(assn),P) = abs_assn(aTP_Lamp_ch(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),P)) ).

% uminus_assn_def
tff(fact_1920_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N5: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N5) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5)) )
       => ~ ! [N5: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N5)) )
             => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5)) ) ) ) ).

% int_cases3
tff(fact_1921_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N2: nat,K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),K))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N2)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_1922_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N2: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N2)),K) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),K)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_1923_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N2) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N2))
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_1924_not__zle__0__negative,axiom,
    ! [N2: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))))) ).

% not_zle_0_negative
tff(fact_1925_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N2)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_1926_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N5: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N5))) ) ).

% negD
tff(fact_1927_negative__zless__0,axiom,
    ! [N2: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_1928_verit__less__mono__div__int2,axiom,
    ! [A5: int,B4: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A5),B4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N2)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N2))) ) ) ).

% verit_less_mono_div_int2
tff(fact_1929_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1930_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,N2: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),N2) = 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))),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) ) ) ) ).

% power_minus'
tff(fact_1931_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_1932_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_1933_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_1934_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_1935_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_1936_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_1937_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_1938_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_1939_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ~ ! [N5: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N5)) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5)) ) ) ).

% neg_int_cases
tff(fact_1940_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_1941_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_1942_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_1943_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_1944_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F3: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N2)) )
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,M)),K))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F3,N2)))
           => ? [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I3))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N2))
                & ( aa(nat,int,F3,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_1945_incr__lemma,axiom,
    ! [D3: int,Z2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z2))),one_one(int))),D3)))) ) ).

% incr_lemma
tff(fact_1946_decr__lemma,axiom,
    ! [D3: int,X: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z2))),one_one(int))),D3))),Z2)) ) ).

% decr_lemma
tff(fact_1947_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R3),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R3)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_1948_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1949_nat__ivt__aux,axiom,
    ! [N2: nat,F3: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F3,N2)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N2))
              & ( aa(nat,int,F3,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1950_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_1951_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1952_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( pp(aa(product_prod(int,int),bool,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,J2: int] :
            ( pp(aa(product_prod(int,int),bool,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),J2)))
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),J2))
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J2)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J2)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_1953_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_1954_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_1955_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_1956_option_Osize_I3_J,axiom,
    ! [A: $tType] : aa(option(A),nat,size_size(option(A)),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(3)
tff(fact_1957_ComplI,axiom,
    ! [A: $tType,C2: A,A5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,C2),A5))
     => pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A5))) ) ).

% ComplI
tff(fact_1958_Compl__iff,axiom,
    ! [A: $tType,C2: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A5)))
    <=> ~ pp(aa(set(A),bool,member(A,C2),A5)) ) ).

% Compl_iff
tff(fact_1959_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(set(A),set(A),uminus_uminus(set(A)),B4) )
    <=> ( A5 = B4 ) ) ).

% Compl_eq_Compl_iff
tff(fact_1960_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_le_zero
tff(fact_1961_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% zero_less_ceiling
tff(fact_1962_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_numeral
tff(fact_1963_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_less_one
tff(fact_1964_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% one_le_ceiling
tff(fact_1965_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_1966_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).

% ceiling_le_one
tff(fact_1967_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).

% one_less_ceiling
tff(fact_1968_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).

% ceiling_less_zero
tff(fact_1969_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).

% zero_le_ceiling
tff(fact_1970_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_1971_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_1972_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_1973_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_1974_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ci(fun(A,bool),fun(A,bool),P)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,bool),set(A),collect(A),P)) ).

% Collect_neg_eq
tff(fact_1975_Compl__eq,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_cj(set(A),fun(A,bool),A5)) ).

% Compl_eq
tff(fact_1976_ComplD,axiom,
    ! [A: $tType,C2: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C2),aa(set(A),set(A),uminus_uminus(set(A)),A5)))
     => ~ pp(aa(set(A),bool,member(A,C2),A5)) ) ).

% ComplD
tff(fact_1977_double__complement,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = A5 ).

% double_complement
tff(fact_1978_uminus__set__def,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5))) ).

% uminus_set_def
tff(fact_1979_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_1980_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_1981_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% ceiling_less_cancel
tff(fact_1982_option_Osize__neq,axiom,
    ! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_1983_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))) ) ).

% ceiling_add_le
tff(fact_1984_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le
tff(fact_1985_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_1986_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N2: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N2)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N2)),A3)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_1987_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N2))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_1988_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N2: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N2)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N2)),A3)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_1989_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N2))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_1990_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_1991_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_1992_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_1993_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2)) ) ) ).

% of_int_le_iff
tff(fact_1994_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).

% of_int_less_iff
tff(fact_1995_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_1996_nat__le__0,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int)))
     => ( aa(int,nat,nat2,Z2) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_1997_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa(int,nat,nat2,I2) = zero_zero(nat) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_1998_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).

% zless_nat_conj
tff(fact_1999_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = Z2 ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = zero_zero(int) ) ) ) ).

% int_nat_eq
tff(fact_2000_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_2001_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) ) ) ).

% of_int_0_le_iff
tff(fact_2002_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_2003_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ) ).

% of_int_0_less_iff
tff(fact_2004_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: num,Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N2)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N2)),Z2)) ) ) ).

% of_int_numeral_le_iff
tff(fact_2005_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int,N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),aa(num,A,numeral_numeral(A),N2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),aa(num,int,numeral_numeral(int),N2))) ) ) ).

% of_int_le_numeral_iff
tff(fact_2006_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: num,Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N2)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),N2)),Z2)) ) ) ).

% of_int_numeral_less_iff
tff(fact_2007_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int,N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),aa(num,A,numeral_numeral(A),N2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),aa(num,int,numeral_numeral(int),N2))) ) ) ).

% of_int_less_numeral_iff
tff(fact_2008_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_2009_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z2)) ) ) ).

% of_int_1_le_iff
tff(fact_2010_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),one_one(int))) ) ) ).

% of_int_less_1_iff
tff(fact_2011_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z2)) ) ) ).

% of_int_1_less_iff
tff(fact_2012_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ) ).

% zero_le_floor
tff(fact_2013_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A))) ) ) ).

% floor_less_zero
tff(fact_2014_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_le_floor
tff(fact_2015_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% zero_less_floor
tff(fact_2016_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ).

% zero_less_nat_eq
tff(fact_2017_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_le_zero
tff(fact_2018_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_numeral
tff(fact_2019_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% one_le_floor
tff(fact_2020_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_less_one
tff(fact_2021_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z2)) = aa(int,A,ring_1_of_int(A),Z2) ) ) ) ).

% of_nat_nat
tff(fact_2022_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2023_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2024_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2))) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2025_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),X)) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2026_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z2)) ) ).

% one_less_nat_eq
tff(fact_2027_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_2028_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_2029_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_2030_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% floor_less_neg_numeral
tff(fact_2031_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2032_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N2: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N2)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2)),A3)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2033_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2034_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N2: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N2)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2)),A3)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2035_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N2)),aa(int,nat,nat2,A3)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2)),A3)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2036_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2037_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N2)),aa(int,nat,nat2,A3)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N2)),A3)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2038_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)) ) ).

% of_int_floor_le
tff(fact_2039_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ) ).

% le_floor_iff
tff(fact_2040_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% floor_less_iff
tff(fact_2041_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_le_of_int
tff(fact_2042_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_less_of_int
tff(fact_2043_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),X)) ) ).

% ex_of_int_less
tff(fact_2044_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),X)) ) ).

% mult_of_int_commute
tff(fact_2045_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs2)))
             => pp(aa(list(A),bool,P,Ys)) )
         => pp(aa(list(A),bool,P,Xs2)) )
     => pp(aa(list(A),bool,P,Xs)) ) ).

% length_induct
tff(fact_2046_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R3)))),R3)) ) ) ).

% of_nat_floor
tff(fact_2047_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z2 ) ) ) ) ).

% floor_unique
tff(fact_2048_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_2049_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T6: A] :
          ( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T6)))
        <=> ! [I4: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),T6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T6),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A)))) )
             => pp(aa(int,bool,P,I4)) ) ) ) ).

% floor_split
tff(fact_2050_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_2051_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_2052_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).

% floor_correct
tff(fact_2053_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))) ) ) ).

% floor_mono
tff(fact_2054_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% floor_less_cancel
tff(fact_2055_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y))) ) ).

% nat_mono
tff(fact_2056_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z7))
       => ( ( aa(int,nat,nat2,Z2) = aa(int,nat,nat2,Z7) )
        <=> ( Z2 = Z7 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_2057_all__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ! [X_12: nat] : pp(aa(nat,bool,P,X_12))
    <=> ! [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
         => pp(aa(nat,bool,P,aa(int,nat,nat2,X4))) ) ) ).

% all_nat
tff(fact_2058_ex__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
    <=> ? [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
          & pp(aa(nat,bool,P,aa(int,nat,nat2,X4))) ) ) ).

% ex_nat
tff(fact_2059_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ).

% le_of_int_ceiling
tff(fact_2060_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X))) ) ).

% floor_le_ceiling
tff(fact_2061_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3)),P3)) ) ) ).

% floor_divide_lower
tff(fact_2062_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K)) ) ) ) ) ).

% of_int_of_nat
tff(fact_2063_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3))) ) ) ).

% floor_divide_upper
tff(fact_2064_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).

% nat_mono_iff
tff(fact_2065_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)))) ) ).

% le_floor_add
tff(fact_2066_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(int,nat,nat2,Z2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z2)) ) ).

% zless_nat_eq_int_zless
tff(fact_2067_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),N2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N2))) ) ).

% nat_le_iff
tff(fact_2068_nat__0__le,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = Z2 ) ) ).

% nat_0_le
tff(fact_2069_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z2 )
    <=> ( ( M = aa(int,nat,nat2,Z2) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) ) ) ).

% int_eq_iff
tff(fact_2070_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R3),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R3))))) ) ).

% of_nat_ceiling
tff(fact_2071_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% ceiling_le_iff
tff(fact_2072_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ) ).

% less_ceiling_iff
tff(fact_2073_nat__plus__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X5),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_plus_as_int
tff(fact_2074_nat__times__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X5),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_times_as_int
tff(fact_2075_nat__minus__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X5),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_minus_as_int
tff(fact_2076_nat__div__as__int,axiom,
    ! [X5: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X5),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_div_as_int
tff(fact_2077_nat__mod__as__int,axiom,
    ! [X5: nat,Xa3: nat] : modulo_modulo(nat,X5,Xa3) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X5),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_mod_as_int
tff(fact_2078_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% of_int_nonneg
tff(fact_2079_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% of_int_pos
tff(fact_2080_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N2))),X))
         => ( ( N2 = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_2081_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N2))),X))
         => ( ( N2 = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% of_int_lessD
tff(fact_2082_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int))))) ) ) ).

% floor_exists
tff(fact_2083_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int)))))
          & ! [Y4: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),X))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int))))) )
             => ( Y4 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_2084_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).

% nat_less_eq_zless
tff(fact_2085_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W2))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2)) ) ) ).

% nat_le_eq_zle
tff(fact_2086_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa(int,nat,nat2,W2) = M )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_2087_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa(int,nat,nat2,W2) )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_2088_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N2)),X)) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2089_split__nat,axiom,
    ! [P: fun(nat,bool),I2: int] :
      ( pp(aa(nat,bool,P,aa(int,nat,nat2,I2)))
    <=> ( ! [N: nat] :
            ( ( I2 = aa(nat,int,semiring_1_of_nat(int),N) )
           => pp(aa(nat,bool,P,N)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
         => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_2090_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(int,nat,nat2,K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N2)),K)) ) ) ).

% le_nat_iff
tff(fact_2091_nat__add__distrib,axiom,
    ! [Z2: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z7))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z7)) ) ) ) ).

% nat_add_distrib
tff(fact_2092_nat__mult__distrib,axiom,
    ! [Z2: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z7)) ) ) ).

% nat_mult_distrib
tff(fact_2093_nat__diff__distrib,axiom,
    ! [Z7: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z7))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z7),Z2))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z7)) ) ) ) ).

% nat_diff_distrib
tff(fact_2094_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_2095_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))) ).

% nat_abs_triangle_ineq
tff(fact_2096_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_2097_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_2098_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => ( aa(int,nat,nat2,modulo_modulo(int,X,Y)) = modulo_modulo(nat,aa(int,nat,nat2,X),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_2099_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).

% ceiling_diff_floor_le_1
tff(fact_2100_nat__power__eq,axiom,
    ! [Z2: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z2),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z2)),N2) ) ) ).

% nat_power_eq
tff(fact_2101_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ).

% le_mult_floor
tff(fact_2102_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).

% ceiling_correct
tff(fact_2103_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2)))
           => ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).

% ceiling_unique
tff(fact_2104_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archimedean_ceiling(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3))) ) ) ) ).

% ceiling_eq_iff
tff(fact_2105_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T6: A] :
          ( pp(aa(int,bool,P,archimedean_ceiling(A,T6)))
        <=> ! [I4: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))),T6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T6),aa(int,A,ring_1_of_int(A),I4))) )
             => pp(aa(int,bool,P,I4)) ) ) ) ).

% ceiling_split
tff(fact_2106_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))) ) ) ).

% ceiling_less_iff
tff(fact_2107_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)) ) ) ).

% le_ceiling_iff
tff(fact_2108_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z2)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_2109_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),M))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).

% nat_less_iff
tff(fact_2110_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int)))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z2))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z7))) ) ) ).

% nat_mult_distrib_neg
tff(fact_2111_nat__abs__int__diff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A3) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_2112_diff__nat__eq__if,axiom,
    ! [Z7: int,Z2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z7),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z7)) = aa(int,nat,nat2,Z2) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z7),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z7)) = if(nat,aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z7)),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z7))) ) ) ) ).

% diff_nat_eq_if
tff(fact_2113_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3))) ) ) ).

% ceiling_divide_upper
tff(fact_2114_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)),P3)) ) ) ).

% ceiling_divide_lower
tff(fact_2115_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_2116_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M6: multiset(A),M9: multiset(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(A),nat,size_size(multiset(A)),M6)),aa(multiset(A),nat,size_size(multiset(A)),M9))),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)),M6),M9)))) ).

% diff_size_le_size_Diff
tff(fact_2117_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( 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)),K3) ) )
        | ? [L2: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),zero_zero(int)) )
            & ( L2 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L2) ) )
        | ? [R4: int,L2: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R4) )
            & ( aa(int,int,sgn_sgn(int),R4) = aa(int,int,sgn_sgn(int),L2) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R4)),aa(int,int,abs_abs(int),L2)))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L2)),R4) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2118_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q5: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22) ) ) )
         => ~ ! [R: int,Q5: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R) )
               => ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),A22) )
                 => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),A22)))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2119_nonempty__has__size,axiom,
    ! [A: $tType,S: multiset(A)] :
      ( ( S != zero_zero(multiset(A)) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S))) ) ).

% nonempty_has_size
tff(fact_2120_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,member(B,A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A3)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_2121_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,sgn_sgn(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_2122_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_greater
tff(fact_2123_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,sgn_sgn(A),B2)) ) ).

% divide_sgn
tff(fact_2124_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_2125_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),archimedean_frac(A,X)))
        <=> ~ pp(aa(set(A),bool,member(A,X),ring_1_Ints(A))) ) ) ).

% frac_gt_0_iff
tff(fact_2126_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_2127_union__diff__assoc,axiom,
    ! [A: $tType,C4: multiset(A),B4: multiset(A),A5: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),B4) = 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)),A5),B4)),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B4),C4)) ) ) ).

% union_diff_assoc
tff(fact_2128_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_mult
tff(fact_2129_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,member(A,B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_mult
tff(fact_2130_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_2131_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).

% abs_mult_sgn
tff(fact_2132_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,abs_abs(A),A3)) = A3 ) ).

% sgn_mult_abs
tff(fact_2133_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),aa(A,A,abs_abs(A),X)) = X ) ).

% mult_sgn_abs
tff(fact_2134_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X))) ) ).

% frac_ge_0
tff(fact_2135_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).

% frac_lt_1
tff(fact_2136_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: A] :
          ( ( archimedean_frac(A,X) = A3 )
        <=> ( pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_2137_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_1_pos
tff(fact_2138_zsgn__def,axiom,
    ! [I2: int] :
      ( ( ( I2 = zero_zero(int) )
       => ( aa(int,int,sgn_sgn(int),I2) = zero_zero(int) ) )
      & ( ( I2 != zero_zero(int) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
           => ( aa(int,int,sgn_sgn(int),I2) = one_one(int) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
           => ( aa(int,int,sgn_sgn(int),I2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).

% zsgn_def
tff(fact_2139_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_2140_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = one_one(A) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).

% sgn_if
tff(fact_2141_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_2142_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
         => ( ( X != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_2143_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_2144_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,member(A,Y),ring_1_Ints(A)))
           => ( ( X = Y )
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_2145_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% frac_eq
tff(fact_2146_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_2147_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q3: int] :
      ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),L) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),L)))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R3) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2148_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,member(B,A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A3)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_2149_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),From),To))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From) ) ) ) ).

% slice_len
tff(fact_2150_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2151_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [N2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N2)),N2) = 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))),N2)),semiring_char_0_fact(A,N2)) ) ).

% pochhammer_same
tff(fact_2152_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( semiring_char_0_fact(A,N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_2153_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( ( M = zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).

% fact_num_eq_if
tff(fact_2154_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2155_cpmi,axiom,
    ! [D4: int,P: fun(int,bool),P6: fun(int,bool),B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z5))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P6,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( pp(aa(set(int),bool,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb2: int] :
                      ( pp(aa(set(int),bool,member(int,Xb2),B4))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa(int,bool,P6,X3))
              <=> pp(aa(int,bool,P6,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
           => ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P6,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa: int] :
                        ( pp(aa(set(int),bool,member(int,Xa),B4))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2156_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L3: A,H3: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L3,H3) )
        <=> ( ( ( L = L3 )
              & ( H = H3 ) )
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
              & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L3),H3)) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_2157_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or1337092689740270186AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% atLeastAtMost_iff
tff(fact_2158_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_2159_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% atLeastatMost_empty_iff
tff(fact_2160_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_2161_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_2162_slice__complete,axiom,
    ! [A: $tType,Xs: list(A)] : slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs),Xs) = Xs ).

% slice_complete
tff(fact_2163_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2))),semiring_char_0_fact(A,N2)) ) ).

% fact_Suc
tff(fact_2164_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D4: multiset(A),C4: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B4),D4))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D4),C4))) ) ) ).

% union_le_mono1
tff(fact_2165_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D4: multiset(A),C4: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B4),D4))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D4))) ) ) ).

% union_le_mono2
tff(fact_2166_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: multiset(A),C4: multiset(A),B4: multiset(A),D4: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),A5),C4))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B4),D4))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D4))) ) ) ) ).

% union_less_mono
tff(fact_2167_mset__distrib,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A),M6: multiset(A),N7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7) )
     => ~ ! [Am: multiset(A),An: multiset(A)] :
            ( ( A5 = 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)] :
                ( ( B4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
               => ( ( M6 = 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_2168_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N2))) ) ).

% fact_mono_nat
tff(fact_2169_fact__ge__self,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),semiring_char_0_fact(nat,N2))) ).

% fact_ge_self
tff(fact_2170_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_2171_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N2))) ) ) ).

% fact_less_mono_nat
tff(fact_2172_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N2))) ) ).

% fact_ge_zero
tff(fact_2173_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N2)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_2174_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N2)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_2175_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N2))) ) ).

% fact_gt_zero
tff(fact_2176_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N2))) ) ).

% fact_ge_1
tff(fact_2177_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N2))) ) ) ).

% fact_mono
tff(fact_2178_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_2179_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_2180_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_2181_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_2182_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A3),K))) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_2183_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
        <=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
              | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
                & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
                  | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3)) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2184_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N2))) ).

% fact_ge_Suc_0_nat
tff(fact_2185_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N2))) ) ) ) ).

% fact_less_mono
tff(fact_2186_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N2),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_2187_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,N2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N2),N2)))) ) ).

% fact_le_power
tff(fact_2188_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% Suc_times_gbinomial
tff(fact_2189_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_2190_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2191_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( ( ( K = zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K) = one_one(A) ) )
          & ( ( K != zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_ck(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K)) ) ) ) ) ).

% gbinomial_code
tff(fact_2192_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),R3)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N2),R3))) ) ).

% fact_div_fact_le_pow
tff(fact_2193_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_2194_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% gbinomial_factors
tff(fact_2195_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A3)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_2196_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N2))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),N2)) ) ).

% gbinomial_index_swap
tff(fact_2197_periodic__finite__ex,axiom,
    ! [D3: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P,X3))
          <=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) )
       => ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
        <=> ? [X4: int] :
              ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D3)))
              & pp(aa(int,bool,P,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2198_aset_I7_J,axiom,
    ! [D4: int,A5: set(int),T6: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),A5))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T6),X5))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T6),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4))) ) ) ) ).

% aset(7)
tff(fact_2199_aset_I5_J,axiom,
    ! [D4: int,T6: int,A5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,T6),A5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X5),T6))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)),T6)) ) ) ) ) ).

% aset(5)
tff(fact_2200_aset_I4_J,axiom,
    ! [D4: int,T6: int,A5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,T6),A5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X5 != T6 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4) != T6 ) ) ) ) ) ).

% aset(4)
tff(fact_2201_aset_I3_J,axiom,
    ! [D4: int,T6: int,A5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T6),one_one(int))),A5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X5 = T6 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4) = T6 ) ) ) ) ) ).

% aset(3)
tff(fact_2202_bset_I7_J,axiom,
    ! [D4: int,T6: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,T6),B4))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B4))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T6),X5))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T6),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4))) ) ) ) ) ).

% bset(7)
tff(fact_2203_bset_I5_J,axiom,
    ! [D4: int,B4: set(int),T6: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),B4))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X5),T6))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)),T6)) ) ) ) ).

% bset(5)
tff(fact_2204_bset_I4_J,axiom,
    ! [D4: int,T6: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,T6),B4))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B4))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X5 != T6 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4) != T6 ) ) ) ) ) ).

% bset(4)
tff(fact_2205_bset_I3_J,axiom,
    ! [D4: int,T6: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T6),one_one(int))),B4))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B4))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X5 = T6 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4) = T6 ) ) ) ) ) ).

% bset(3)
tff(fact_2206_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_2207_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2208_aset_I8_J,axiom,
    ! [D4: int,A5: set(int),T6: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),A5))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T6),X5))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T6),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4))) ) ) ) ).

% aset(8)
tff(fact_2209_aset_I6_J,axiom,
    ! [D4: int,T6: int,A5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T6),one_one(int))),A5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X5),T6))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)),T6)) ) ) ) ) ).

% aset(6)
tff(fact_2210_bset_I8_J,axiom,
    ! [D4: int,T6: int,B4: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T6),one_one(int))),B4))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B4))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T6),X5))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T6),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4))) ) ) ) ) ).

% bset(8)
tff(fact_2211_bset_I6_J,axiom,
    ! [D4: int,B4: set(int),T6: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),B4))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X5),T6))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)),T6)) ) ) ) ).

% bset(6)
tff(fact_2212_cppi,axiom,
    ! [D4: int,P: fun(int,bool),P6: fun(int,bool),A5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),X3))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P6,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( pp(aa(set(int),bool,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb2: int] :
                      ( pp(aa(set(int),bool,member(int,Xb2),A5))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa(int,bool,P6,X3))
              <=> pp(aa(int,bool,P6,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
           => ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P6,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa: int] :
                        ( pp(aa(set(int),bool,member(int,Xa),A5))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2213_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From)))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_2214_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cl(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cl(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cl(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_2215_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cm(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cm(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cm(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_2216_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cn(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cn(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_cn(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_2217_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_co(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_co(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_co(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_2218_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,C2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_cp(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_cp(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2)) ) ) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_cp(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_2219_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_2220_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),X: B,A5: set(B)] :
      ( ( B2 = aa(B,A,F3,X) )
     => ( pp(aa(set(B),bool,member(B,X),A5))
       => pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),A5))) ) ) ).

% image_eqI
tff(fact_2221_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cq(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_2222_image__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_2223_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F3),A5) )
    <=> ( A5 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_2224_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A5) = bot_bot(set(A)) )
    <=> ( A5 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_2225_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_2226_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cr(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I2,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_2227_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cs(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_2228_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,bool),F3: fun(B,A),G3: 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_ct(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F3),G3)),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,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,bool),set(B),collect(B),P)))),aa(set(B),set(A),image2(B,A,G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_cu(fun(B,bool),fun(B,bool),P))))) ).

% if_image_distrib
tff(fact_2229_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
         => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_2230_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_cv(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_2231_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,X: A,A5: set(A),B2: B,F3: fun(A,B)] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ( ( B2 = aa(A,B,F3,X) )
       => pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ).

% rev_image_eqI
tff(fact_2232_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(set(B),set(A),image2(B,A,F3),A5)))
         => pp(aa(A,bool,P,X3)) )
     => ! [X5: B] :
          ( pp(aa(set(B),bool,member(B,X5),A5))
         => pp(aa(A,bool,P,aa(B,A,F3,X5))) ) ) ).

% ball_imageD
tff(fact_2233_image__cong,axiom,
    ! [B: $tType,A: $tType,M6: set(A),N7: set(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( M6 = N7 )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),N7))
           => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) )
       => ( aa(set(A),set(B),image2(A,B,F3),M6) = aa(set(A),set(B),image2(A,B,G3),N7) ) ) ) ).

% image_cong
tff(fact_2234_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),aa(set(B),set(A),image2(B,A,F3),A5)))
          & pp(aa(A,bool,P,X5)) )
     => ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A5))
          & pp(aa(A,bool,P,aa(B,A,F3,X3))) ) ) ).

% bex_imageD
tff(fact_2235_image__iff,axiom,
    ! [A: $tType,B: $tType,Z2: A,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,Z2),aa(set(B),set(A),image2(B,A,F3),A5)))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
          & ( Z2 = aa(B,A,F3,X4) ) ) ) ).

% image_iff
tff(fact_2236_imageI,axiom,
    ! [B: $tType,A: $tType,X: A,A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => pp(aa(set(B),bool,member(B,aa(A,B,F3,X)),aa(set(A),set(B),image2(A,B,F3),A5))) ) ).

% imageI
tff(fact_2237_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),A5)))
     => ~ ! [X3: B] :
            ( ( B2 = aa(B,A,F3,X3) )
           => ~ pp(aa(set(B),bool,member(B,X3),A5)) ) ) ).

% imageE
tff(fact_2238_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),A5: set(C)] : aa(set(B),set(A),image2(B,A,F3),aa(set(C),set(B),image2(C,B,G3),A5)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_cw(fun(B,A),fun(fun(C,B),fun(C,A)),F3),G3)),A5) ).

% image_image
tff(fact_2239_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),P: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_cx(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F3),A5),P)) = aa(set(B),set(A),image2(B,A,F3),aa(fun(B,bool),set(B),collect(B),aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_cy(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F3),A5),P))) ).

% Compr_image_eq
tff(fact_2240_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K2: nat,M5: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M5) ).

% prod_decode_aux.cases
tff(fact_2241_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
         => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),N7),M6)) ) ) ).

% mset_le_asym
tff(fact_2242_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K5: multiset(A),M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),K5),M6))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),K5),N7)) ) ) ) ).

% mset_le_trans
tff(fact_2243_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A)] : ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),M6)) ) ).

% mset_le_irrefl
tff(fact_2244_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
         => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),N7),M6)) ) ) ).

% mset_le_not_sym
tff(fact_2245_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A)] : ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),M6)) ) ).

% mset_le_not_refl
tff(fact_2246_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less_eq(multiset(A)),M6),N7))
        <=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
            | ( M6 = N7 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_2247_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image2(B,A,F3),A5)))
    <=> ? [AA: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),AA),A5))
          & ( B4 = aa(set(B),set(A),image2(B,A,F3),AA) ) ) ) ).

% subset_image_iff
tff(fact_2248_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A5)),B4))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
         => pp(aa(set(A),bool,member(A,aa(B,A,F3,X4)),B4)) ) ) ).

% image_subset_iff
tff(fact_2249_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image2(B,A,F3),A5)))
     => ~ ! [C6: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C6),A5))
           => ( B4 != aa(set(B),set(A),image2(B,A,F3),C6) ) ) ) ).

% subset_imageE
tff(fact_2250_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,B),B4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(set(B),bool,member(B,aa(A,B,F3,X3)),B4)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B4)) ) ).

% image_subsetI
tff(fact_2251_image__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),aa(set(A),set(B),image2(A,B,F3),B4))) ) ).

% image_mono
tff(fact_2252_image__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B4: set(B)] : aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = 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,F3),A5)),aa(set(B),set(A),image2(B,A,F3),B4)) ).

% image_Un
tff(fact_2253_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: fun(A,B),B4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(set(B),bool,member(B,aa(A,B,F3,X3)),B4)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(fun(A,bool),set(A),collect(A),P))),B4)) ) ).

% image_Collect_subsetI
tff(fact_2254_obtain__list__from__elements,axiom,
    ! [A: $tType,N2: nat,P: fun(A,fun(nat,bool))] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N2))
         => ? [Li: A] : pp(aa(nat,bool,aa(A,fun(nat,bool),P,Li),I3)) )
     => ~ ! [L4: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L4) = N2 )
           => ~ ! [I: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N2))
                 => pp(aa(nat,bool,aa(A,fun(nat,bool),P,aa(nat,A,nth(A,L4),I)),I)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2255_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2256_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,bool))] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
         => ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,I4),X_12)) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
             => pp(aa(A,bool,aa(nat,fun(A,bool),P,I4),aa(nat,A,nth(A,Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_2257_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2258_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B4: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))),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,F3),A5)),aa(set(B),set(A),image2(B,A,F3),B4)))) ).

% image_Int_subset
tff(fact_2259_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B4: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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,F3),A5)),aa(set(B),set(A),image2(B,A,F3),B4))),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4)))) ).

% image_diff_subset
tff(fact_2260_ex__nat__less,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ? [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N2))
          & pp(aa(nat,bool,P,M3)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)))
          & pp(aa(nat,bool,P,X4)) ) ) ).

% ex_nat_less
tff(fact_2261_all__nat__less,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N2))
         => pp(aa(nat,bool,P,M3)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)))
         => pp(aa(nat,bool,P,X4)) ) ) ).

% all_nat_less
tff(fact_2262_None__notin__image__Some,axiom,
    ! [A: $tType,A5: set(A)] : ~ pp(aa(set(option(A)),bool,member(option(A),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A5))) ).

% None_notin_image_Some
tff(fact_2263_nth__rotate1,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),N2) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,N2),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_2264_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_2265_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_2266_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_2267_product__nth,axiom,
    ! [A: $tType,B: $tType,N2: nat,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys2)),N2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(list(B),nat,size_size(list(B)),Ys2)))),aa(nat,B,nth(B,Ys2),modulo_modulo(nat,N2,aa(list(B),nat,size_size(list(B)),Ys2)))) ) ) ).

% product_nth
tff(fact_2268_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T6: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cz(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T6)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_cz(A,fun(A,A),A3)),T6)) ) ).

% translation_subtract_Compl
tff(fact_2269_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T6: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cz(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T6)) = 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_cz(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_cz(A,fun(A,A),A3)),T6)) ) ).

% translation_subtract_diff
tff(fact_2270_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T6: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cz(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T6)) = 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_cz(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_cz(A,fun(A,A),A3)),T6)) ) ).

% translation_subtract_Int
tff(fact_2271_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N2: nat,M: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( ( ( X = one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2)) = 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),N2),one_one(nat))),M)) ) )
              & ( ( X != one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_2272_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F3: fun(B,nat),A5: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_da(fun(B,nat),fun(B,A),F3)),A5) ) ).

% of_nat_sum
tff(fact_2273_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F3: fun(B,int),A5: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_db(fun(B,int),fun(B,A),F3)),A5) ) ).

% of_int_sum
tff(fact_2274_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N2: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))),aa(nat,A,G3,aa(nat,nat,suc,N2))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_2275_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A3: A,A5: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_dc(fun(B,A),fun(A,fun(B,A)),F3),A3)),A5),A3) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5),A3) ) ).

% mod_sum_eq
tff(fact_2276_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2277_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,fun(nat,A)),G3),K)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2278_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,N2,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_df(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N2),M)),set_or1337092689740270186AtMost(nat,N2,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2279_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,aa(nat,nat,suc,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2280_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N2))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2281_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))),aa(nat,A,G3,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,N2))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2282_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N2: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dg(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,aa(nat,nat,suc,N2))),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_2283_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I5: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dh(A,fun(nat,fun(nat,A)),X),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),I5)) ) ).

% sum_power_add
tff(fact_2284_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_di(fun(nat,A),fun(nat,fun(A,A)),F3),A3,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_2285_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A),P3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2286_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N2: nat,F3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dj(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M)),aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dj(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,N2)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_2287_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N2: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,N2)),aa(nat,A,F3,M)) ) ) ) ).

% sum_telescope''
tff(fact_2288_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N2: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2289_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dl(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2290_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N2: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_offset
tff(fact_2291_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T6: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S2)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T6)) ) ).

% translation_Int
tff(fact_2292_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A5: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_dm(fun(A,B),fun(A,B),F3)),A5))) ) ).

% sum_abs_ge_zero
tff(fact_2293_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A5: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_dm(fun(A,B),fun(A,B),F3)),A5))) ) ).

% sum_abs
tff(fact_2294_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_2295_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,member(A,I3),I5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I3))) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( pp(aa(set(A),bool,member(A,I3),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I3)),B2))),Delta)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_dn(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2296_abs__sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A5: set(A)] : aa(B,B,abs_abs(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_dm(fun(A,B),fun(A,B),F3)),A5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_dm(fun(A,B),fun(A,B),F3)),A5) ) ).

% abs_sum_abs
tff(fact_2297_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_do(B,A)),A5) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2298_int__sum,axiom,
    ! [B: $tType,F3: fun(B,nat),A5: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F3),A5)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),aTP_Lamp_dp(fun(B,nat),fun(B,int),F3)),A5) ).

% int_sum
tff(fact_2299_sum__subtractf__nat,axiom,
    ! [A: $tType,A5: set(A),G3: fun(A,nat),F3: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G3,X3)),aa(A,nat,F3,X3))) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_dq(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G3),F3)),A5) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G3),A5)) ) ) ).

% sum_subtractf_nat
tff(fact_2300_sum__SucD,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A),N2: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) = aa(nat,nat,suc,N2) )
     => ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X3))) ) ) ).

% sum_SucD
tff(fact_2301_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,fun(C,A)),B4: set(C),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aTP_Lamp_dr(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B4)),A5) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(set(B),fun(C,A),aTP_Lamp_dt(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G3),A5)),B4) ) ).

% sum.swap
tff(fact_2302_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),K5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),K5))) ) ) ).

% sum_mono
tff(fact_2303_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),H: fun(B,A),A5: 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_du(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),A5)) ) ).

% sum.distrib
tff(fact_2304_sum__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_0(B)
     => ! [F3: fun(A,B),A5: set(A),G3: fun(C,B),B4: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),B4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_dw(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F3),G3),B4)),A5) ) ).

% sum_product
tff(fact_2305_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),A5: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),R3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_dx(fun(B,A),fun(A,fun(B,A)),F3),R3)),A5) ) ).

% sum_distrib_right
tff(fact_2306_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R3: A,F3: fun(B,A),A5: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)) = 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_dy(A,fun(fun(B,A),fun(B,A)),R3),F3)),A5) ) ).

% sum_distrib_left
tff(fact_2307_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A5: 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_dz(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A5) = 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),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5)) ) ).

% sum_subtractf
tff(fact_2308_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ea(fun(B,A),fun(B,A),F3)),A5) = aa(A,A,uminus_uminus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)) ) ).

% sum_negf
tff(fact_2309_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),A5: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),R3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_eb(fun(B,A),fun(A,fun(B,A)),F3),R3)),A5) ) ).

% sum_divide_distrib
tff(fact_2310_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5))) ) ) ).

% sum_nonneg
tff(fact_2311_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_2312_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ec(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ed(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_2313_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp0
tff(fact_2314_divmod__nat__def,axiom,
    ! [M: nat,N2: nat] : divmod_nat(M,N2) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2)),modulo_modulo(nat,M,N2)) ).

% divmod_nat_def
tff(fact_2315_sorted__in__between,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: nat,J: nat,L: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),I2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
             => ( sorted_wrt(A,ord_less_eq(A),L)
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I2)),X))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,nth(A,L),J)))
                   => ~ ! [K2: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K2))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),J))
                           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),K2)),X))
                             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat))))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_2316_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ee(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_2317_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ec(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ef(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_2318_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_atMost(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),K)) ) ) ).

% atMost_iff
tff(fact_2319_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_eg(B,A)),A5) = one_one(A) ) ).

% prod.neutral_const
tff(fact_2320_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,nat),A5: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_eh(fun(B,nat),fun(B,A),F3)),A5) ) ).

% of_nat_prod
tff(fact_2321_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F3: fun(B,int),A5: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ei(fun(B,int),fun(B,A),F3)),A5) ) ).

% of_int_prod
tff(fact_2322_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_2323_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),X)),aa(A,set(A),set_ord_atMost(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% atMost_subset_iff
tff(fact_2324_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H3)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H),H3)) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_2325_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N2))),aa(nat,A,G3,aa(nat,nat,suc,N2))) ) ).

% prod.atMost_Suc
tff(fact_2326_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N2: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))),aa(nat,A,G3,aa(nat,nat,suc,N2))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_2327_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,fun(C,A)),B4: set(C),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_ej(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B4)),A5) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(set(B),fun(C,A),aTP_Lamp_el(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G3),A5)),B4) ) ).

% prod.swap
tff(fact_2328_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_em(A,fun(A,bool)),Xs) ).

% sorted_wrt_true
tff(fact_2329_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_2330_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),H: fun(B,A),A5: 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_en(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A5)) ) ).

% prod.distrib
tff(fact_2331_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A5: 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_eo(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A5) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)) ) ).

% prod_dividef
tff(fact_2332_prod__power__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [F3: fun(A,B),A5: set(A),N2: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),N2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(nat,fun(A,B),aTP_Lamp_ep(fun(A,B),fun(nat,fun(A,B)),F3),N2)),A5) ) ).

% prod_power_distrib
tff(fact_2333_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A3: A,A5: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aTP_Lamp_dc(fun(B,A),fun(A,fun(B,A)),F3),A3)),A5),A3) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5),A3) ) ).

% mod_prod_eq
tff(fact_2334_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_eq(fun(B,A),fun(B,A),F3)),A5) ) ).

% abs_prod
tff(fact_2335_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_er(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N2))) ) ).

% prod.atMost_Suc_shift
tff(fact_2336_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_es(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_2337_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% strict_sorted_imp_sorted
tff(fact_2338_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5))) ) ) ).

% prod_nonneg
tff(fact_2339_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5))) ) ) ).

% prod_mono
tff(fact_2340_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5))) ) ) ).

% prod_pos
tff(fact_2341_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5))) ) ) ).

% prod_ge_1
tff(fact_2342_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I2: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Ns)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,nth(nat,Ns),I2))) ) ) ).

% sorted_wrt_less_idx
tff(fact_2343_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L3: A,H3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L3,H3))) ) ).

% not_Iic_le_Icc
tff(fact_2344_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_er(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2345_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F3: fun(B,nat),A5: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,nat),fun(B,A),aTP_Lamp_et(A,fun(fun(B,nat),fun(B,A)),C2),F3)),A5) ) ).

% power_sum
tff(fact_2346_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_eu(fun(nat,A),fun(nat,fun(nat,A)),G3),K)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_2347_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),one_one(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_2348_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_2349_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),I2: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_2350_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_2351_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,N2,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N2),M)),set_or1337092689740270186AtMost(nat,N2,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_2352_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P3))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ew(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ex(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_2353_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_2354_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_2355_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))),aa(nat,A,G3,aa(nat,nat,suc,N2))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2356_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,aa(nat,nat,suc,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_2357_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N2))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_2358_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))),aa(nat,A,G3,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_er(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,N2))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2359_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N2))) ) ).

% sum.atMost_Suc_shift
tff(fact_2360_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),I2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ey(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_atMost(nat),I2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,aa(nat,nat,suc,I2))) ) ).

% sum_telescope
tff(fact_2361_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] : semiring_char_0_fact(A,N2) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ez(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N2))) ) ).

% fact_prod
tff(fact_2362_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F3: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F3),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fa(fun(nat,A),fun(nat,fun(A,A)),F3),A3,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_2363_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4)))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_2364_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).

% sorted_nth_mono
tff(fact_2365_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_2366_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A),P3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_2367_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fb(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N2))),one_one(A))),N2) ) ).

% gbinomial_parallel_sum
tff(fact_2368_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N2)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ez(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N2),M))) ) ) ).

% fact_eq_fact_times
tff(fact_2369_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))) ) ).

% sum_gp_basic
tff(fact_2370_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N2: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fc(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) ) ).

% pochhammer_Suc_prod
tff(fact_2371_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N2: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)))) ) ) ) ).

% sum_power_shift
tff(fact_2372_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N2: nat] : comm_s3205402744901411588hammer(A,A3,N2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fd(A,fun(nat,fun(nat,A)),A3),N2)),set_or1337092689740270186AtMost(nat,one_one(nat),N2)) ) ).

% pochhammer_prod_rev
tff(fact_2373_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N2)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ez(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_2374_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fe(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_2375_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N2: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fd(A,fun(nat,fun(nat,A)),A3),N2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_2376_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P3))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ff(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fg(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2377_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fh(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2378_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N2: nat] :
          ( ( N2 != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fi(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2379_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z2: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N2))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fj(A,fun(nat,A),Z2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_2380_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A5: set(nat),C2: fun(nat,A),D3: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
              & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5)) )
           => ( 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_fk(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A5) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D3,zero_zero(nat))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
                & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5)) )
           => ( 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_fk(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A5) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_2381_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N2,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2382_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ee(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_2383_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit0,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2)) ) ).

% semiring_norm(78)
tff(fact_2384_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit0,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N2)) ) ).

% semiring_norm(71)
tff(fact_2385_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),one2)) ).

% semiring_norm(75)
tff(fact_2386_semiring__norm_I68_J,axiom,
    ! [N2: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),one2),N2)) ).

% semiring_norm(68)
tff(fact_2387_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or7035219750837199246ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% atLeastLessThan_iff
tff(fact_2388_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_2389_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Icc_iff
tff(fact_2390_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_2391_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_2392_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ico_iff
tff(fact_2393_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,J: A,M: A,N2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I2,J)),set_or7035219750837199246ssThan(A,M,N2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),I2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),I2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),N2)) ) ) ) ) ).

% ivl_subset
tff(fact_2394_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,N2: A,M: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),N2))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I2,M)),set_or7035219750837199246ssThan(A,I2,N2)) = set_or7035219750837199246ssThan(A,N2,M) ) ) ) ).

% ivl_diff
tff(fact_2395_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa(nat,nat,binomial(N2),K) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),K)) ) ).

% binomial_eq_0_iff
tff(fact_2396_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),one2)) ).

% semiring_norm(69)
tff(fact_2397_semiring__norm_I76_J,axiom,
    ! [N2: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit0,N2))) ).

% semiring_norm(76)
tff(fact_2398_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fl(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fl(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_2399_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fm(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fm(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_2400_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cr(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I2,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_2401_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fn(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fn(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = one_one(A) ) ) ) ) ) ).

% prod.delta'
tff(fact_2402_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fo(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( 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_fo(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = one_one(A) ) ) ) ) ) ).

% prod.delta
tff(fact_2403_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N2),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) ) ).

% zero_less_binomial_iff
tff(fact_2404_prod__pos__nat__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A5)))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_2405_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2406_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2407_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N2)),one_one(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N2),one2)) ) ) ).

% numeral_le_one_iff
tff(fact_2408_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N2)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N2)) ) ) ).

% one_less_numeral_iff
tff(fact_2409_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_2410_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2411_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_2412_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N2: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))),aa(nat,A,G3,N2)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_2413_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N2: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))),aa(nat,A,G3,N2)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_2414_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A5: set(nat),C2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
              & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fp(fun(nat,A),fun(nat,A),C2)),A5) = aa(nat,A,C2,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
                & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fp(fun(nat,A),fun(nat,A),C2)),A5) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_2415_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) ) ) ).

% one_less_floor
tff(fact_2416_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% floor_le_one
tff(fact_2417_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2418_int__prod,axiom,
    ! [B: $tType,F3: fun(B,nat),A5: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F3),A5)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),aTP_Lamp_dp(fun(B,nat),fun(B,int),F3)),A5) ).

% int_prod
tff(fact_2419_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S: set(A),S4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),bool),aTP_Lamp_fq(set(A),fun(set(A),fun(set(A),bool)),S),S4)))) ) ).

% finite_if_eq_beyond_finite
tff(fact_2420_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B2))) ) ) ).

% infinite_Ico
tff(fact_2421_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N2),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ).

% binomial_le_pow2
tff(fact_2422_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
           => ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
            <=> ( ( A3 = C2 )
                & ( B2 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_2423_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
             => ( A3 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_2424_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
             => ( B2 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_2425_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set(nat),N2: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N7),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N7)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_2426_choose__row__sum,axiom,
    ! [N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2) ).

% choose_row_sum
tff(fact_2427_finite__nat__set__iff__bounded,axiom,
    ! [N7: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N7))
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),M3)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_2428_bounded__nat__set__is__finite,axiom,
    ! [N7: set(nat),N2: nat] :
      ( ! [X3: nat] :
          ( pp(aa(set(nat),bool,member(nat,X3),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),N2)) )
     => pp(aa(set(nat),bool,finite_finite2(nat),N7)) ) ).

% bounded_nat_set_is_finite
tff(fact_2429_finite__nat__set__iff__bounded__le,axiom,
    ! [N7: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N7))
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),M3)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_2430_choose__square__sum,axiom,
    ! [N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fr(nat,fun(nat,nat),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)),N2) ).

% choose_square_sum
tff(fact_2431_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).

% mult_2
tff(fact_2432_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).

% mult_2_right
tff(fact_2433_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),B2) ) ).

% left_add_twice
tff(fact_2434_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),X),one2))
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_2435_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_2436_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% power2_eq_square
tff(fact_2437_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,bool),I2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_fs(fun(nat,bool),fun(nat,fun(nat,bool)),P),I2)))) ).

% finite_M_bounded_by_nat
tff(fact_2438_finite__less__ub,axiom,
    ! [F3: fun(nat,nat),U: nat] :
      ( ! [N5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),aa(nat,nat,F3,N5)))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ft(fun(nat,nat),fun(nat,fun(nat,bool)),F3),U)))) ) ).

% finite_less_ub
tff(fact_2439_sum_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(C),G3: fun(B,fun(C,A)),R2: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(C),bool,finite_finite2(C),B4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_fv(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B4),G3),R2)),A5) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_fx(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A5),G3),R2)),B4) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2440_prod_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(C),G3: fun(B,fun(C,A)),R2: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(C),bool,finite_finite2(C),B4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_fy(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B4),G3),R2)),A5) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_fz(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A5),G3),R2)),B4) ) ) ) ) ).

% prod.swap_restrict
tff(fact_2441_less__exp,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ).

% less_exp
tff(fact_2442_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M))) ) ).

% self_le_ge2_pow
tff(fact_2443_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% power2_nat_le_eq_le
tff(fact_2444_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% power2_nat_le_imp_le
tff(fact_2445_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_2446_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ez(nat,nat)),set_or7035219750837199246ssThan(nat,M,N2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_2447_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),K))
     => ( aa(nat,nat,binomial(N2),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2448_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% half_gt_zero_iff
tff(fact_2449_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% half_gt_zero
tff(fact_2450_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_2451_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% zero_le_power2
tff(fact_2452_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2453_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% power2_le_imp_le
tff(fact_2454_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_2455_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
     => ( aa(nat,nat,binomial(N2),K) = aa(nat,nat,binomial(N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)) ) ) ).

% binomial_symmetric
tff(fact_2456_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_2457_ex__nat__less__eq,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ? [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
          & pp(aa(nat,bool,P,M3)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))
          & pp(aa(nat,bool,P,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_2458_all__nat__less__eq,axiom,
    ! [N2: nat,P: fun(nat,bool)] :
      ( ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
         => pp(aa(nat,bool,P,M3)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))
         => pp(aa(nat,bool,P,X4)) ) ) ).

% all_nat_less_eq
tff(fact_2459_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> ( ( N2 = zero_zero(nat) )
        | ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2460_less__2__cases,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
     => ( ( N2 = zero_zero(nat) )
        | ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2461_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_2462_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_le_square_iff
tff(fact_2463_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
                & ~ ? [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),S))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa3),X3)) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_2464_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X6: set(A)] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),X6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Xa3)) ) )
           => ~ pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).

% infinite_growing
tff(fact_2465_binomial__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N2),R3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N2),R3))) ) ).

% binomial_le_pow
tff(fact_2466_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N2)))) ) ).

% diff_le_diff_pow
tff(fact_2467_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ).

% binomial_r_part_sum
tff(fact_2468_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_2469_choose__linear__sum,axiom,
    ! [N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ga(nat,fun(nat,nat),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)))) ).

% choose_linear_sum
tff(fact_2470_sum__mono__inv,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F3: fun(I6,A),I5: set(I6),G3: fun(I6,A),I2: I6] :
          ( ( aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),F3),I5) = aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),G3),I5) )
         => ( ! [I3: I6] :
                ( pp(aa(set(I6),bool,member(I6,I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,I3)),aa(I6,A,G3,I3))) )
           => ( pp(aa(set(I6),bool,member(I6,I2),I5))
             => ( pp(aa(set(I6),bool,finite_finite2(I6),I5))
               => ( aa(I6,A,F3,I2) = aa(I6,A,G3,I2) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_2471_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B2))) ) ) ).

% infinite_Icc
tff(fact_2472_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% mult_numeral_1_right
tff(fact_2473_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).

% mult_numeral_1
tff(fact_2474_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gb(nat,fun(nat,nat),M)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,nat,binomial(aa(nat,nat,suc,N2)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_2475_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,M,N2)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_2476_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,fun(nat,A)),G3),K)),set_or7035219750837199246ssThan(nat,M,N2)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_2477_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_er(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,M,N2)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_2478_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_gd(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_2479_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_eu(fun(nat,A),fun(nat,fun(nat,A)),G3),K)),set_or7035219750837199246ssThan(nat,M,N2)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_2480_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2))
           => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) = modulo_modulo(A,A3,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2481_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_gf(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_2482_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% power2_less_imp_less
tff(fact_2483_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2484_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_2485_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2486_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_2487_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N2: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N2)),aa(num,A,numeral_numeral(A),N2)) ) ).

% numeral_code(2)
tff(fact_2488_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G3: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_gg(set(B),fun(fun(B,bool),fun(B,bool)),A5),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_gh(fun(B,A),fun(fun(B,bool),fun(B,A)),G3),P)),A5) ) ) ) ).

% sum.inter_filter
tff(fact_2489_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_sum
tff(fact_2490_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2491_filter__preserves__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_gi(fun(A,nat),fun(A,bool),M6))))
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_gj(fun(A,nat),fun(fun(A,bool),fun(A,bool)),M6),P)))) ) ).

% filter_preserves_multiset
tff(fact_2492_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) ) ).

% zero_le_even_power'
tff(fact_2493_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2494_sum_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),H: fun(B,A),G3: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_gl(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S),H),G3)),aa(set(B),set(C),image2(B,C,G3),S)) ) ) ) ).

% sum.image_gen
tff(fact_2495_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2496_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_less_1
tff(fact_2497_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G3: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_gg(set(B),fun(fun(B,bool),fun(B,bool)),A5),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_gm(fun(B,A),fun(fun(B,bool),fun(B,A)),G3),P)),A5) ) ) ) ).

% prod.inter_filter
tff(fact_2498_prod_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),H: fun(B,A),G3: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_gn(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S),H),G3)),aa(set(B),set(C),image2(B,C,G3),S)) ) ) ) ).

% prod.image_gen
tff(fact_2499_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,N2: 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)),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)) ) ).

% minus_power_mult_self
tff(fact_2500_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) = 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),N2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% power_odd_eq
tff(fact_2501_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2502_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% div_2_gt_zero
tff(fact_2503_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_go(A,fun(A,fun(A,bool)),A3),B2)))) ) ).

% finite_int_segment
tff(fact_2504_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N2),K))) ) ).

% zero_less_binomial
tff(fact_2505_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A3: B,C2: B,B2: B,D3: B,G3: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C2 )
         => ( ( B2 = D3 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),set_or7035219750837199246ssThan(B,C2,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_2506_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A3: B,C2: B,B2: B,D3: B,G3: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C2 )
         => ( ( B2 = D3 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),set_or7035219750837199246ssThan(B,C2,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_2507_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N2),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N2),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% choose_mult
tff(fact_2508_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_2509_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N2),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gp(nat,fun(nat,fun(nat,A)),K),N2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_2510_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2511_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_2512_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,P3: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),P3))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,N2,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_2513_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N2: nat,P3: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),P3))
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,N2,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_2514_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_diff
tff(fact_2515_binomial__code,axiom,
    ! [N2: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),K))
       => ( aa(nat,nat,binomial(N2),K) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),K))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
           => ( aa(nat,nat,binomial(N2),K) = aa(nat,nat,binomial(N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
           => ( aa(nat,nat,binomial(N2),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)),one_one(nat)),N2,one_one(nat))),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_2516_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,P3: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,N2,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_2517_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2518_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_2519_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5) = zero_zero(A) )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A5))
                 => ( aa(B,A,F3,X4) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2520_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),T6: set(C),G3: fun(C,A),I2: fun(C,B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(C),bool,finite_finite2(C),T6))
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,member(C,X3),T6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G3,X3))) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S2))
                   => ? [Xa3: C] :
                        ( pp(aa(set(C),bool,member(C,Xa3),T6))
                        & ( aa(C,B,I2,Xa3) = X3 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(C,A,G3,Xa3))) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),S2)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G3),T6))) ) ) ) ) ) ).

% sum_le_included
tff(fact_2521_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A5: set(I6),F3: fun(I6,A),G3: fun(I6,A)] :
          ( pp(aa(set(I6),bool,finite_finite2(I6),A5))
         => ( ! [X3: I6] :
                ( pp(aa(set(I6),bool,member(I6,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,X3)),aa(I6,A,G3,X3))) )
           => ( ? [X5: I6] :
                  ( pp(aa(set(I6),bool,member(I6,X5),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I6,A,F3,X5)),aa(I6,A,G3,X5))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),F3),A5)),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),G3),A5))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_2522_sum__strict__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( A5 != bot_bot(set(B)) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5))) ) ) ) ) ).

% sum_strict_mono
tff(fact_2523_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R2: fun(A,fun(A,bool)),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,one_one(A)),one_one(A)))
         => ( ! [X12: A,Y12: A,X22: A,Y22: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X12),X22))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R2,Y12),Y22)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),times_times(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),times_times(A),X22),Y22))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H,X3)),aa(B,A,G3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S))) ) ) ) ) ) ).

% prod.related
tff(fact_2524_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
       => ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N5)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_2525_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
       => ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N5)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_2526_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( set_or7035219750837199246ssThan(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I2,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_2527_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N2: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2))) ) ).

% mask_eq_sum_exp
tff(fact_2528_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(1)
tff(fact_2529_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(2)
tff(fact_2530_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gq(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% sum.in_pairs
tff(fact_2531_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2))) ).

% mask_eq_sum_exp_nat
tff(fact_2532_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gr(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% prod.in_pairs
tff(fact_2533_gauss__sum__nat,axiom,
    ! [N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ez(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,suc,N2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% gauss_sum_nat
tff(fact_2534_prod__int__eq,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_gs(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_2535_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gq(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).

% sum.in_pairs_0
tff(fact_2536_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gr(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).

% prod.in_pairs_0
tff(fact_2537_sum__choose__lower,axiom,
    ! [R3: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gt(nat,fun(nat,nat),R3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R3),N2))),N2) ).

% sum_choose_lower
tff(fact_2538_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gu(nat,fun(nat,nat),N2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_2539_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gu(nat,fun(nat,nat),N2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_2540_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),S2) = zero_zero(A) )
             => ( pp(aa(set(B),bool,member(B,I2),S2))
               => ( aa(B,A,F3,I2) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2541_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F3: fun(B,A),B4: A,I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),S2) = B4 )
             => ( pp(aa(set(B),bool,member(B,I2),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),B4)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2542_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2543_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2) = modulo_modulo(A,A3,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2544_add__mset__in__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),A3: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_gi(fun(A,nat),fun(A,bool),M6))))
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_gv(fun(A,nat),fun(A,fun(A,bool)),M6),A3)))) ) ).

% add_mset_in_multiset
tff(fact_2545_sum_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G3: fun(B,A),B4: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(B),fun(B,A),aTP_Lamp_gw(fun(B,A),fun(set(B),fun(B,A)),G3),B4)),A5) ) ) ) ).

% sum.inter_restrict
tff(fact_2546_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T2: set(C),G3: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T2))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G3),S)),T2))
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_gx(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G3),H)),T2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S) ) ) ) ) ) ).

% sum.group
tff(fact_2547_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_gy(fun(B,A),fun(B,bool),G3)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2548_diff__preserves__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),N7: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_gi(fun(A,nat),fun(A,bool),M6))))
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,nat),fun(A,bool),aTP_Lamp_gz(fun(A,nat),fun(fun(A,nat),fun(A,bool)),M6),N7)))) ) ).

% diff_preserves_multiset
tff(fact_2549_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T2: set(C),G3: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T2))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G3),S)),T2))
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_ha(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G3),H)),T2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S) ) ) ) ) ) ).

% prod.group
tff(fact_2550_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set(nat),N2: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N7),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N7)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2551_prod_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G3: fun(B,A),B4: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(B),fun(B,A),aTP_Lamp_hb(fun(B,A),fun(set(B),fun(B,A)),G3),B4)),A5) ) ) ) ).

% prod.inter_restrict
tff(fact_2552_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z2: A,W2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))) ) ).

% power_numeral_even
tff(fact_2553_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_hc(fun(B,A),fun(B,bool),G3)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_2554_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_hd(A,fun(A,bool),A3)))) ) ).

% finite_abs_int_segment
tff(fact_2555_neg__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_2556_pos__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_2557_pos__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,B2,A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_2558_arith__series__nat,axiom,
    ! [A3: nat,D3: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_he(nat,fun(nat,fun(nat,nat)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),D3)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_2559_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_2560_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_2561_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_2562_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ez(nat,nat)),set_or1337092689740270186AtMost(nat,M,N2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_2563_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N2))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_2564_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_2565_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))),aa(nat,A,G3,N2)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_2566_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N2))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_2567_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_2568_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,N2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))) ) ) ) ).

% sum.last_plus
tff(fact_2569_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,N2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))) ) ) ) ).

% prod.last_plus
tff(fact_2570_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),I2: B,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( pp(aa(set(B),bool,member(B,I2),I5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I2)))
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,member(B,I3),I5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),I5))) ) ) ) ) ) ).

% sum_pos2
tff(fact_2571_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,member(B,I3),I5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),I5))) ) ) ) ) ).

% sum_pos
tff(fact_2572_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( pp(aa(set(A),bool,member(A,I2),I5))
           => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F3,I2)))
             => ( ! [I3: A] :
                    ( pp(aa(set(A),bool,member(A,I3),I5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,I3))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I5))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_2573_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( pp(aa(set(A),bool,member(A,I3),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F3,I3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I5))) ) ) ) ) ).

% less_1_prod
tff(fact_2574_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)))),aa(nat,nat,binomial(N2),K)) = semiring_char_0_fact(nat,N2) ) ) ).

% binomial_fact_lemma
tff(fact_2575_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C4: set(B),A5: set(B),B4: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C4))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,member(B,A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A5)))
                   => ( aa(B,A,G3,A6) = zero_zero(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B4)))
                     => ( aa(B,A,H,B5) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B4) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),C4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C4) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2576_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C4: set(B),A5: set(B),B4: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C4))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,member(B,A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A5)))
                   => ( aa(B,A,G3,A6) = zero_zero(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B4)))
                     => ( aa(B,A,H,B5) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),C4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C4) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2577_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T2: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),T2) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2578_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T2: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),T2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2579_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T2: set(B),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,H,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2580_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T2: set(B),S: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),T2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2581_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: set(B),A5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),A5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2582_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A5: set(B),B4: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4)) = 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),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),B4)) ) ) ) ) ).

% sum_diff
tff(fact_2583_sum_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T2: set(B),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,finite_finite2(B),S))
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,H,I3) = zero_zero(A) ) )
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),T2)))
                   => ( aa(B,A,G3,I3) = zero_zero(A) ) )
               => ( ! [X3: B] :
                      ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T2)))
                     => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T2) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_2584_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B4: set(B),A5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),A5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B4)) ) ) ) ) ).

% prod.subset_diff
tff(fact_2585_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C4: set(B),A5: set(B),B4: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C4))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,member(B,A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A5)))
                   => ( aa(B,A,G3,A6) = one_one(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B4)))
                     => ( aa(B,A,H,B5) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B4) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),C4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C4) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_2586_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C4: set(B),A5: set(B),B4: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C4))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,member(B,A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A5)))
                   => ( aa(B,A,G3,A6) = one_one(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B4)))
                     => ( aa(B,A,H,B5) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),C4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C4) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_2587_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T2: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),T2) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_2588_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T2: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),T2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_2589_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T2: set(B),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,H,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_2590_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T2: set(B),S: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),T2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_2591_sum_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),B4)) ) ) ) ) ).

% sum.union_inter
tff(fact_2592_sum_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G3: fun(B,A),B4: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4))) ) ) ) ).

% sum.Int_Diff
tff(fact_2593_prod_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B4)) ) ) ) ) ).

% prod.union_inter
tff(fact_2594_prod_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G3: fun(B,A),B4: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4))) ) ) ) ).

% prod.Int_Diff
tff(fact_2595_prod_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T2: set(B),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T2))
         => ( pp(aa(set(B),bool,finite_finite2(B),S))
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
                 => ( aa(B,A,H,I3) = one_one(A) ) )
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),T2)))
                   => ( aa(B,A,G3,I3) = one_one(A) ) )
               => ( ! [X3: B] :
                      ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T2)))
                     => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T2) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_2596_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_gs(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)))) ).

% prod_int_plus_eq
tff(fact_2597_neg__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A3))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2598_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D3: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_hf(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))) = 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),N2)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),D3))) ) ).

% double_arith_series
tff(fact_2599_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_2600_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_hg(nat,fun(nat,fun(nat,nat)),M),N2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N2)),M) ) ) ).

% sum_choose_diagonal
tff(fact_2601_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N2: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dg(fun(nat,A),fun(nat,A),F3)),set_or7035219750837199246ssThan(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,N2)),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_2602_sum__diff__nat,axiom,
    ! [A: $tType,B4: set(A),A5: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_2603_vandermonde,axiom,
    ! [M: nat,N2: nat,R3: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hh(nat,fun(nat,fun(nat,fun(nat,nat))),M),N2),R3)),aa(nat,set(nat),set_ord_atMost(nat),R3)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)),R3) ).

% vandermonde
tff(fact_2604_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,N2,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hi(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N2),M)),set_or7035219750837199246ssThan(nat,N2,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_2605_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R3)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2606_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hj(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hl(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)) ) ).

% sum.nested_swap
tff(fact_2607_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,N2,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N2),M)),set_or7035219750837199246ssThan(nat,N2,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_2608_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hn(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)) ) ).

% prod.nested_swap
tff(fact_2609_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: nat] : semiring_char_0_fact(A,N2) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),N2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) ) ).

% fact_prod_rev
tff(fact_2610_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hq(nat,fun(nat,A),M)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_2611_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2612_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N2))
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_gs(int,int)),set_or1337092689740270186AtMost(int,M,N2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(int,int,aa(int,fun(int,int),plus_plus(int),N2),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_2613_sum_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),P: fun(B,bool),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_hr(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G3)),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(fun(B,bool),set(B),collect(B),P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P))))) ) ) ) ).

% sum.If_cases
tff(fact_2614_prod_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),P: fun(B,bool),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_hs(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G3)),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(fun(B,bool),set(B),collect(B),P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P))))) ) ) ) ).

% prod.If_cases
tff(fact_2615_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N2: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2))),semiring_char_0_fact(A,N2)) ) ).

% fact_double
tff(fact_2616_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N2),K)))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2617_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2618_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( aa(nat,nat,binomial(N2),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2619_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D3: A,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ht(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),D3)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_2620_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N2: 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),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_2621_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2622_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N2: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))),aa(nat,A,G3,N2)) ) ) ) ) ).

% sum.head_if
tff(fact_2623_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),zero_zero(int)))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R3)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_2624_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N2: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2))),aa(nat,A,G3,N2)) ) ) ) ) ).

% prod.head_if
tff(fact_2625_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),A5))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) ) )
           => ( ( A5 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5))) ) ) ) ) ).

% prod_mono_strict
tff(fact_2626_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B4: set(B),A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
           => ( ! [B5: B] :
                  ( pp(aa(set(B),bool,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A5)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,B5))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),B4))) ) ) ) ) ).

% sum_mono2
tff(fact_2627_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
     => ( aa(nat,nat,binomial(N2),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_2628_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),B4)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_2629_sum__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A5: set(B),B4: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) ) ) ) ) ).

% sum_Un
tff(fact_2630_sum_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4) = bot_bot(set(B)) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),B4)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_2631_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B4)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_2632_prod_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4) = bot_bot(set(B)) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B4)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_2633_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B4: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = 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),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ) ).

% sum_Un2
tff(fact_2634_sum_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A5)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) ) ) ) ) ).

% sum.union_diff2
tff(fact_2635_prod_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A5)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) ) ) ) ) ).

% prod.union_diff2
tff(fact_2636_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z2: A,N2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))),comm_s3205402744901411588hammer(A,Z2,N2))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N2)) ) ).

% pochhammer_double
tff(fact_2637_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N2: 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)),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2638_sum__Un__nat,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),B4))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_2639_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,N2,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_df(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N2),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N2),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2640_binomial,axiom,
    ! [A3: nat,B2: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),N2) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hu(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ).

% binomial
tff(fact_2641_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,N2,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N2),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N2),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2642_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hv(A,fun(nat,A),R3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R3),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_2643_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N2: nat] : comm_s3205402744901411588hammer(A,A3,N2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fc(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)) ) ).

% pochhammer_prod
tff(fact_2644_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,nat,binomial(N2),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2645_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N2),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N2)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)))) ) ) ) ).

% binomial_fact
tff(fact_2646_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N2),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K))) ) ) ) ).

% fact_binomial
tff(fact_2647_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => ( semiring_char_0_fact(A,N2) = 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),N2),K),N2)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K))) ) ) ) ).

% fact_split
tff(fact_2648_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_2649_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A5: set(A),B2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
           => ( pp(aa(set(A),bool,member(A,B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)))
             => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F3,B2)))
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,member(A,X3),B4))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3))) )
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B4))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2650_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A5: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
           => ( ! [B5: A] :
                  ( pp(aa(set(A),bool,member(A,B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,B5))) )
             => ( ! [A6: A] :
                    ( pp(aa(set(A),bool,member(A,A6),A5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,A6))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B4))) ) ) ) ) ) ).

% prod_mono2
tff(fact_2651_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A5: set(B),B4: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)))
                 => ( aa(B,A,F3,X3) != zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) ) ) ) ) ) ).

% prod_Un
tff(fact_2652_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hv(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_2653_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N2) = 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_hw(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).

% binomial_ring
tff(fact_2654_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B2: A,N2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),N2) = 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_hx(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).

% pochhammer_binomial_sum
tff(fact_2655_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hy(A,fun(nat,fun(nat,A)),A3),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_2656_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),semiring_char_0_fact(A,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hz(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_2657_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A3),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hz(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_2658_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% half_negative_int_iff
tff(fact_2659_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% half_nonnegative_int_iff
tff(fact_2660_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R3: A,Q3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R3),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),R3) ) ) ) ) ).

% divmod_step_eq
tff(fact_2661_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2662_nat__bit__induct,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N5: nat] :
            ( pp(aa(nat,bool,P,N5))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5))) ) )
       => ( ! [N5: nat] :
              ( pp(aa(nat,bool,P,N5))
             => pp(aa(nat,bool,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5)))) )
         => pp(aa(nat,bool,P,N2)) ) ) ) ).

% nat_bit_induct
tff(fact_2663_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ca(nat,fun(nat,bool)),K)))) ).

% finite_Collect_le_nat
tff(fact_2664_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
        | pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),Q))) )
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ag(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2665_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ai(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
        & pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2666_finite__interval__int1,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ia(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int1
tff(fact_2667_finite__interval__int4,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ib(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int4
tff(fact_2668_of__nat__id,axiom,
    ! [N2: nat] : aa(nat,nat,semiring_1_of_nat(nat),N2) = N2 ).

% of_nat_id
tff(fact_2669_finite__Int,axiom,
    ! [A: $tType,F4: set(A),G4: set(A)] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),F4))
        | pp(aa(set(A),bool,finite_finite2(A),G4)) )
     => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F4),G4))) ) ).

% finite_Int
tff(fact_2670_finite__Un,axiom,
    ! [A: $tType,F4: set(A),G4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G4)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),F4))
        & pp(aa(set(A),bool,finite_finite2(A),G4)) ) ) ).

% finite_Un
tff(fact_2671_finite__interval__int3,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ic(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int3
tff(fact_2672_finite__interval__int2,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_id(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int2
tff(fact_2673_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),K)))) ).

% finite_Collect_less_nat
tff(fact_2674_finite__Collect__subsets,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ie(set(A),fun(set(A),bool),A5)))) ) ).

% finite_Collect_subsets
tff(fact_2675_finite__maxlen,axiom,
    ! [A: $tType,M6: set(list(A))] :
      ( pp(aa(set(list(A)),bool,finite_finite2(list(A)),M6))
     => ? [N5: nat] :
        ! [X5: list(A)] :
          ( pp(aa(set(list(A)),bool,member(list(A),X5),M6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X5)),N5)) ) ) ).

% finite_maxlen
tff(fact_2676_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_if(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_2677_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ).

% not_finite_existsD
tff(fact_2678_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),R2: fun(A,fun(B,bool))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A5))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,member(B,Xa3),B4))
                  & pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Xa3)) ) )
         => ? [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),B4))
              & ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ig(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),A5),R2),X3)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_2679_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                & ! [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),A5))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3))
                     => ( X3 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2680_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3))
                & ! [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),A5))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa3),X3))
                     => ( X3 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2681_finite_OemptyI,axiom,
    ! [A: $tType] : pp(aa(set(A),bool,finite_finite2(A),bot_bot(set(A)))) ).

% finite.emptyI
tff(fact_2682_infinite__imp__nonempty,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
     => ( S != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_2683_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P: fun(set(A),bool)] :
      ( ! [B7: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),aa(set(B),set(A),image2(B,A,F3),A5)))
         => pp(aa(set(A),bool,P,B7)) )
    <=> ! [B7: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),A5))
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F3),B7))) ) ) ).

% all_subset_image
tff(fact_2684_finite__subset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% finite_subset
tff(fact_2685_infinite__super,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T2))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
       => ~ pp(aa(set(A),bool,finite_finite2(A),T2)) ) ) ).

% infinite_super
tff(fact_2686_rev__finite__subset,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
       => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% rev_finite_subset
tff(fact_2687_finite__UnI,axiom,
    ! [A: $tType,F4: set(A),G4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,finite_finite2(A),G4))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G4))) ) ) ).

% finite_UnI
tff(fact_2688_Un__infinite,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
     => ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2))) ) ).

% Un_infinite
tff(fact_2689_infinite__Un,axiom,
    ! [A: $tType,S: set(A),T2: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)))
    <=> ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
        | ~ pp(aa(set(A),bool,finite_finite2(A),T2)) ) ) ).

% infinite_Un
tff(fact_2690_finite__psubset__induct,axiom,
    ! [A: $tType,A5: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [A11: set(A)] :
            ( pp(aa(set(A),bool,finite_finite2(A),A11))
           => ( ! [B8: set(A)] :
                  ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B8),A11))
                 => pp(aa(set(A),bool,P,B8)) )
             => pp(aa(set(A),bool,P,A11)) ) )
       => pp(aa(set(A),bool,P,A5)) ) ) ).

% finite_psubset_induct
tff(fact_2691_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),image2(A,B,F3),A5)))
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
            & ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ih(set(A),fun(fun(A,B),fun(A,fun(A,bool))),A5),F3),X3)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_2692_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A5: set(A),F3: fun(nat,A),N2: nat] :
      ( ( A5 = aa(set(nat),set(A),image2(nat,A,F3),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2))) )
     => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ).

% nat_seg_image_imp_finite
tff(fact_2693_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
    <=> ? [N: nat,F10: fun(nat,A)] : A5 = aa(set(nat),set(A),image2(nat,A,F10),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N))) ) ).

% finite_conv_nat_seg_image
tff(fact_2694_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
                & ! [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),A5))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa3),X3))
                     => ( X3 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_2695_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
                & ! [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),A5))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3))
                     => ( X3 = Xa3 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_2696_finite__surj,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),image2(A,B,F3),A5)))
       => pp(aa(set(B),bool,finite_finite2(B),B4)) ) ) ).

% finite_surj
tff(fact_2697_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image2(B,A,F3),A5)))
       => ? [C6: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C6),A5))
            & pp(aa(set(B),bool,finite_finite2(B),C6))
            & ( B4 = aa(set(B),set(A),image2(B,A,F3),C6) ) ) ) ) ).

% finite_subset_image
tff(fact_2698_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P: fun(set(A),bool)] :
      ( ? [B7: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B7))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),aa(set(B),set(A),image2(B,A,F3),A5)))
          & pp(aa(set(A),bool,P,B7)) )
    <=> ? [B7: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B7))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),A5))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F3),B7))) ) ) ).

% ex_finite_subset_image
tff(fact_2699_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P: fun(set(A),bool)] :
      ( ! [B7: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),B7))
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),aa(set(B),set(A),image2(B,A,F3),A5))) )
         => pp(aa(set(A),bool,P,B7)) )
    <=> ! [B7: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),B7))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),A5)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F3),B7))) ) ) ).

% all_finite_subset_image
tff(fact_2700_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)),zero_zero(int))) ).

% not_exp_less_eq_0_int
tff(fact_2701_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_2702_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_2703_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5668285175392031749et_bit(A,N2,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_2704_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,N2,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_2705_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2638667681897837118et_bit(A,N2,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_2706_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N2: nat,A3: A] :
          ( ( ( N2 = zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N2),A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) )
          & ( ( N2 != zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_2707_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2638667681897837118et_bit(int,N2,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2708_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5668285175392031749et_bit(int,N2,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2709_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,N2,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2710_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2638667681897837118et_bit(int,N2,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% unset_bit_negative_int_iff
tff(fact_2711_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5668285175392031749et_bit(int,N2,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% set_bit_negative_int_iff
tff(fact_2712_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se8732182000553998342ip_bit(int,N2,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% flip_bit_negative_int_iff
tff(fact_2713_unset__bit__less__eq,axiom,
    ! [N2: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se2638667681897837118et_bit(int,N2,K)),K)) ).

% unset_bit_less_eq
tff(fact_2714_set__bit__greater__eq,axiom,
    ! [K: int,N2: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),bit_se5668285175392031749et_bit(int,N2,K))) ).

% set_bit_greater_eq
tff(fact_2715_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ).

% signed_take_bit_int_less_exp
tff(fact_2716_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2717_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2718_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2719_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2720_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2721_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)),K))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N2))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2722_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2723_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2724_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2725_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N2: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_2726_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_2727_finite__int__iff__bounded__le,axiom,
    ! [S: set(int)] :
      ( pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S)),aa(int,set(int),set_ord_atMost(int),K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_2728_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ii(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2) ) ) ) ).

% choose_odd_sum
tff(fact_2729_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ij(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2) ) ) ) ).

% choose_even_sum
tff(fact_2730_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_2731_finite__nat__iff__bounded__le,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_atMost(nat),K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_2732_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),one_one(nat)))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_2733_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( C2 = zero_zero(A) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2734_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( C2 = zero_zero(A) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2735_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2736_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2737_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2738_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2739_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ).

% unit_prod
tff(fact_2740_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_2741_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),A3) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_2742_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_2743_dvd__1__left,axiom,
    ! [K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K)) ).

% dvd_1_left
tff(fact_2744_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),A3) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_2745_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) ) ) ) ).

% unit_mult_div_div
tff(fact_2746_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N2: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2747_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) ) ) ) ).

% even_mult_iff
tff(fact_2748_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ) ).

% even_power
tff(fact_2749_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2750_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2751_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_2752_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2))) ) ) ).

% even_diff_nat
tff(fact_2753_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_2754_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2755_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2)))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2756_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2757_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2758_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N2),M))
       => ( M = N2 ) ) ) ).

% dvd_antisym
tff(fact_2759_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ? [B9: A,C7: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B9),C7) )
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B9),B2))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C7),C2)) ) ) ) ).

% division_decomp
tff(fact_2760_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
         => ~ ! [X3: A,Y3: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y3) )
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),A3))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y3),B2)) ) ) ) ) ).

% dvd_productE
tff(fact_2761_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_2762_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A3: A,B2: A,K: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% dvdI
tff(fact_2763_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
        <=> ? [K3: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvd_def
tff(fact_2764_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult
tff(fact_2765_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult2
tff(fact_2766_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ).

% dvd_mult_left
tff(fact_2767_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ).

% dvd_triv_left
tff(fact_2768_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ).

% mult_dvd_mono
tff(fact_2769_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ).

% dvd_mult_right
tff(fact_2770_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))) ) ).

% dvd_triv_right
tff(fact_2771_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2))) ) ) ).

% dvd_diff_nat
tff(fact_2772_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ik(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ik(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_2773_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ik(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ik(A,fun(A,bool),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_2774_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X5))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% pinf(9)
tff(fact_2775_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X5))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% pinf(10)
tff(fact_2776_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X5),Z3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% minf(9)
tff(fact_2777_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X5),Z3))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% minf(10)
tff(fact_2778_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_2779_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_2780_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2781_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2782_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2783_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2784_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).

% is_unit_mult_iff
tff(fact_2785_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,D3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),C2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2786_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ).

% dvd_mult_imp_div
tff(fact_2787_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2788_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_2789_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_2790_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_2791_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).

% dvd_pos_nat
tff(fact_2792_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
       => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N2),M)) ) ) ).

% nat_dvd_not_less
tff(fact_2793_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N2: nat,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M))) ) ) ) ).

% dvd_power_le
tff(fact_2794_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N2: nat,B2: A,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),B2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),B2)) ) ) ) ).

% power_le_dvd
tff(fact_2795_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ).

% le_imp_power_dvd
tff(fact_2796_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N2)) ) ) ).

% dvd_minus_self
tff(fact_2797_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N2))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_2798_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N2)) ) ) ) ).

% dvd_diffD1
tff(fact_2799_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M)) ) ) ) ).

% dvd_diffD
tff(fact_2800_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N2: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_char_0_fact(A,N2)),semiring_char_0_fact(A,M))) ) ) ).

% fact_dvd
tff(fact_2801_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,bool),L: A] :
          ( ? [X4: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4)))
        <=> ? [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A))))
              & pp(aa(A,bool,P,X4)) ) ) ) ).

% unity_coeff_ex
tff(fact_2802_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ).

% unit_dvdE
tff(fact_2803_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D4: A,T6: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D4))
         => ! [X5: A,K4: A] :
              ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T6)))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T6))) ) ) ) ).

% inf_period(4)
tff(fact_2804_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D4: A,T6: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D4))
         => ! [X5: A,K4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T6)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T6))) ) ) ) ).

% inf_period(3)
tff(fact_2805_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),D3))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D3),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2806_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2807_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2808_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2809_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2810_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2811_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_2812_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2813_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2814_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = C2 )
          <=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_2815_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y))) ) ) ).

% round_mono
tff(fact_2816_prod__dvd__prod__subset,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B4: set(B),A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),B4))) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_2817_prod__dvd__prod__subset2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [B4: set(B),A5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
           => ( ! [A6: B] :
                  ( pp(aa(set(B),bool,member(B,A6),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F3,A6)),aa(B,A,G3,A6))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B4))) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_2818_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) ) ) ).

% dvd_imp_le
tff(fact_2819_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N2)) ) ) ).

% dvd_mult_cancel
tff(fact_2820_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N2)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2821_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,N2))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2)))) ) ).

% fact_fact_dvd_fact
tff(fact_2822_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,N2)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N2),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2823_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N2,Q3) )
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2824_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X))) ) ).

% floor_le_round
tff(fact_2825_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X))) ) ).

% ceiling_ge_round
tff(fact_2826_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),semiring_char_0_fact(nat,N2))) ) ) ).

% dvd_fact
tff(fact_2827_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_il(nat,fun(nat,bool),M)))) ) ).

% finite_divisors_nat
tff(fact_2828_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_im(nat,fun(nat,bool)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_2829_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2830_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2831_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B5: A] :
                  ( ( B5 != zero_zero(A) )
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B5),one_one(A)))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B5 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B5) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B5) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B5) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2832_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ~ ! [B5: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B5) ) ) ).

% evenE
tff(fact_2833_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,N2: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A)))
              | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ) ) ).

% dvd_power_iff
tff(fact_2834_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N2: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
            | ( X = one_one(A) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2))) ) ) ).

% dvd_power
tff(fact_2835_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),M))
      <=> ( N2 = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2836_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),M)),M))
      <=> ( N2 = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2837_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),K)))),semiring_char_0_fact(A,N2))) ) ) ).

% choose_dvd
tff(fact_2838_dvd__minus__add,axiom,
    ! [Q3: nat,N2: nat,R3: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q3),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),Q3)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)),Q3)))) ) ) ) ).

% dvd_minus_add
tff(fact_2839_mod__nat__eqI,axiom,
    ! [R3: nat,N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R3),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),R3)))
         => ( modulo_modulo(nat,M,N2) = R3 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2840_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),N2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ).

% power_dvd_imp_le
tff(fact_2841_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A3 ) ) ) ).

% even_two_times_div_two
tff(fact_2842_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: nat,A3: A,B2: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2))) ) ) ) ).

% power_mono_odd
tff(fact_2843_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).

% odd_pos
tff(fact_2844_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ).

% dvd_power_iff_le
tff(fact_2845_sum__div__partition,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A5: set(B),F3: fun(B,A),B2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_in(fun(B,A),fun(A,fun(B,A)),F3),B2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_io(fun(B,A),fun(A,fun(B,bool)),F3),B2))))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_ip(fun(B,A),fun(A,fun(B,bool)),F3),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_2846_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: A,M: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z2))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),aa(int,A,ring_1_of_int(A),M))))) ) ).

% round_diff_minimal
tff(fact_2847_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))
         => ~ ! [B5: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B5)),one_one(A)) ) ) ).

% oddE
tff(fact_2848_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2))) ) ) ).

% zero_le_even_power
tff(fact_2849_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: nat,A3: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% zero_le_odd_power
tff(fact_2850_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2851_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N2))) ) ) ) ).

% power_mono_even
tff(fact_2852_finite__transitivity__chain,axiom,
    ! [A: $tType,A5: set(A),R2: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [X3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),X3))
       => ( ! [X3: A,Y3: A,Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,Y3),Z3))
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Z3)) ) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => ? [Y4: A] :
                    ( pp(aa(set(A),bool,member(A,Y4),A5))
                    & pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y4)) ) )
           => ( A5 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_2853_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)))
        <=> ( ( N2 = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2854_unbounded__k__infinite,axiom,
    ! [K: nat,S: set(nat)] :
      ( ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),M5))
         => ? [N8: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N8))
              & pp(aa(set(nat),bool,member(nat,N8),S)) ) )
     => ~ pp(aa(set(nat),bool,finite_finite2(nat),S)) ) ).

% unbounded_k_infinite
tff(fact_2855_infinite__nat__iff__unbounded,axiom,
    ! [S: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ! [M3: nat] :
        ? [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
          & pp(aa(set(nat),bool,member(nat,N),S)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_2856_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ! [M3: nat] :
        ? [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N))
          & pp(aa(set(nat),bool,member(nat,N),S)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_2857_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2))))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ).

% even_mask_div_iff'
tff(fact_2858_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2859_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2))))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2) = zero_zero(A) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ) ).

% even_mask_div_iff
tff(fact_2860_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2) = zero_zero(A) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2861_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% of_int_round_le
tff(fact_2862_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_2863_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_2864_infinite__int__iff__unbounded__le,axiom,
    ! [S: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ! [M3: int] :
        ? [N: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M3),aa(int,int,abs_abs(int),N)))
          & pp(aa(set(int),bool,member(int,N),S)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_2865_infinite__int__iff__unbounded,axiom,
    ! [S: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ! [M3: int] :
        ? [N: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M3),aa(int,int,abs_abs(int),N)))
          & pp(aa(set(int),bool,member(int,N),S)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_2866_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% of_int_round_abs_le
tff(fact_2867_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
         => ( archimedean_round(A,X) = N2 ) ) ) ).

% round_unique'
tff(fact_2868_in__finite__psubset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A5),B4)),finite_psubset(A)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
        & pp(aa(set(A),bool,finite_finite2(A),B4)) ) ) ).

% in_finite_psubset
tff(fact_2869_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_2870_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_iq(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_2871_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] :
          ( ( ( N2 = zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3) = zero_zero(A) ) )
          & ( ( N2 != zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% take_bit_rec
tff(fact_2872_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_ir(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_2873_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N2: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).

% one_mod_2_pow_eq
tff(fact_2874_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( pp(P)
           => pp(Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_2875_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( ~ pp(P)
            & pp(Q) ) ) ) ).

% of_bool_less_iff
tff(fact_2876_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% of_nat_of_bool
tff(fact_2877_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A5: set(product_prod(A,B)),F3: fun(A,fun(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),A5))
     => pp(aa(set(C),bool,member(C,aa(B,C,aa(A,fun(B,C),F3,A3),B2)),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)),A5))) ) ).

% pair_imageI
tff(fact_2878_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F3,A3),B2) ).

% case_prod_conv
tff(fact_2879_curry__case__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : product_curry(A,B,C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)) = F3 ).

% curry_case_prod
tff(fact_2880_case__prod__curry,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),product_curry(A,B,C,F3)) = F3 ).

% case_prod_curry
tff(fact_2881_take__bit__of__Suc__0,axiom,
    ! [N2: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ).

% take_bit_of_Suc_0
tff(fact_2882_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P)))
        <=> pp(P) ) ) ).

% zero_less_of_bool_iff
tff(fact_2883_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A)))
        <=> ~ pp(P) ) ) ).

% of_bool_less_one_iff
tff(fact_2884_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_2885_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% sgn_mult_self_eq
tff(fact_2886_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N2),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).

% take_bit_of_1
tff(fact_2887_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N2: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).

% sgn_of_nat
tff(fact_2888_sum__mult__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A5: set(B),F3: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_is(fun(B,A),fun(fun(B,bool),fun(B,A)),F3),P)),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(fun(B,bool),set(B),collect(B),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_2889_sum__of__bool__mult__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A5: set(B),P: fun(B,bool),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( 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_it(fun(B,bool),fun(fun(B,A),fun(B,A)),P),F3)),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(fun(B,bool),set(B),collect(B),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_2890_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N2: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) ) ).

% take_bit_of_exp
tff(fact_2891_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N2: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_of_2
tff(fact_2892_nested__case__prod__simp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F3),X),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_iu(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F3),Y)),X) ).

% nested_case_prod_simp
tff(fact_2893_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F3),X),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_iu(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F3),Y)),X) ).

% case_prod_app
tff(fact_2894_prod_Ocase__distrib,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,H: fun(C,D),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(C,D,H,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)) = aa(product_prod(A,B),D,aa(fun(A,fun(B,D)),fun(product_prod(A,B),D),product_case_prod(A,B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_iv(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),H),F3)),Prod) ).

% prod.case_distrib
tff(fact_2895_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,N2),B2) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_2896_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q3)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),Q3))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2897_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_2898_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fconj(P,Q)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_conj
tff(fact_2899_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C)),P3: product_prod(A,B)] :
      ( ! [X3: A,Y3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) = Q3 )
         => ( aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(B,C,aa(A,fun(B,C),G3,X3),Y3) ) )
     => ( ( P3 = Q3 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3),Q3) ) ) ) ).

% split_cong
tff(fact_2900_old_Oprod_Ocase,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,fun(B,C)),X1: A,X2: B] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2)) = aa(B,C,aa(A,fun(B,C),F3,X1),X2) ).

% old.prod.case
tff(fact_2901_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P3) = P3 ).

% case_prod_Pair_iden
tff(fact_2902_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)) ) ) ).

% take_bit_nat_eq
tff(fact_2903_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_2904_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),G3: fun(product_prod(A,B),C)] :
      ( ! [X3: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(product_prod(A,B),C,G3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) = G3 ) ) ).

% cond_case_prod_eta
tff(fact_2905_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_iw(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3)) = F3 ).

% case_prod_eta
tff(fact_2906_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,bool),P: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
      ( pp(aa(A,bool,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z2)))
     => ~ ! [X3: B,Y3: C] :
            ( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y3) )
           => ~ pp(aa(A,bool,Q,aa(C,A,aa(B,fun(C,A),P,X3),Y3))) ) ) ).

% case_prodE2
tff(fact_2907_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P))) ) ).

% zero_less_eq_of_bool
tff(fact_2908_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A))) ) ).

% of_bool_less_eq_one
tff(fact_2909_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2910_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K))) ).

% take_bit_nonnegative
tff(fact_2911_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_2912_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)),zero_zero(int))) ).

% not_take_bit_negative
tff(fact_2913_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% take_bit_int_greater_self_iff
tff(fact_2914_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N2: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3)) = aa(A,A,if(fun(A,A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M),bit_se2584673776208193580ke_bit(A,N2),bit_ri4674362597316999326ke_bit(A,M)),A3) ) ).

% signed_take_bit_take_bit
tff(fact_2915_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,M: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),bit_se2638667681897837118et_bit(A,M,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),bit_se2638667681897837118et_bit(A,M,A3)) = bit_se2638667681897837118et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3)) ) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_2916_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,M: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),bit_se5668285175392031749et_bit(A,M,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),bit_se5668285175392031749et_bit(A,M,A3)) = bit_se5668285175392031749et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3)) ) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_2917_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,M: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),bit_se8732182000553998342ip_bit(A,M,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N2),bit_se8732182000553998342ip_bit(A,M,A3)) = bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3)) ) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_2918_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),N2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N2),M))
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2919_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N2))
       => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N2),M)) ) ) ).

% zdvd_not_zless
tff(fact_2920_uncurry__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : uncurry(A,B,C,F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) ).

% uncurry_def
tff(fact_2921_internal__case__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] : produc5280177257484947105e_prod(A,B,C) = product_case_prod(A,B,C) ).

% internal_case_prod_def
tff(fact_2922_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero(int) )
     => pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aTP_Lamp_ix(int,fun(int,bool),I2)))) ) ).

% finite_divisors_int
tff(fact_2923_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2)))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N2),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2924_zdvd__imp__le,axiom,
    ! [Z2: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z2),N2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),N2)) ) ) ).

% zdvd_imp_le
tff(fact_2925_dvd__imp__le__int,axiom,
    ! [I2: int,D3: int] :
      ( ( I2 != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),I2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),D3)),aa(int,int,abs_abs(int),I2))) ) ) ).

% dvd_imp_le_int
tff(fact_2926_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(N2)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% subset_decode_imp_le
tff(fact_2927_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( ( L = zero_zero(int) )
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) )
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L)) ) ) ).

% mod_int_pos_iff
tff(fact_2928_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N2: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N2),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_2929_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_2930_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ).

% take_bit_nat_less_exp
tff(fact_2931_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),M) = M )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2932_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ).

% take_bit_int_less_exp
tff(fact_2933_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),M)),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_2934_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,bool),A3: A] :
          ( ! [A6: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A6),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A6 )
             => pp(aa(A,bool,P,A6)) )
         => ( ! [A6: A,B5: bool] :
                ( pp(aa(A,bool,P,A6))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A6))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A6 )
                 => pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A6)))) ) )
           => pp(aa(A,bool,P,A3)) ) ) ) ).

% bits_induct
tff(fact_2935_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2936_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_2937_divmod__nat__if,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( ( N2 = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) )
       => ( divmod_nat(M,N2) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ) )
      & ( ~ ( ( N2 = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) )
       => ( divmod_nat(M,N2) = 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_iy(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2),N2)) ) ) ) ).

% divmod_nat_if
tff(fact_2938_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,Z2)),M))
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z2),aa(nat,int,semiring_1_of_nat(int),M))) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_2939_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2940_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2941_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N2: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ).

% exp_mod_exp
tff(fact_2942_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% take_bit_Suc
tff(fact_2943_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))) ) ) ).

% take_bit_int_less_eq
tff(fact_2944_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))),aa(int,int,bit_se2584673776208193580ke_bit(int,N2),K))) ) ).

% take_bit_int_greater_eq
tff(fact_2945_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(int,nat,nat2,K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)) ) ) ).

% even_nat_iff
tff(fact_2946_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_iz(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_2947_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N2: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2))) ) ).

% exp_div_exp_eq
tff(fact_2948_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_ja(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_2949_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N2),aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2950_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))) = case_option(int,num,zero_zero(int),aTP_Lamp_jb(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N2)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_2951_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N2: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,N2)) = 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_jc(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N2)) ) ).

% divmod_algorithm_code(5)
tff(fact_2952_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R3: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R3),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_2953_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N2: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2))
           => ( unique8689654367752047608divmod(A,M,N2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2))
           => ( unique8689654367752047608divmod(A,M,N2) = unique1321980374590559556d_step(A,N2,unique8689654367752047608divmod(A,M,aa(num,num,bit0,N2))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_2954_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N2: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,N2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_2955_case__prodI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),F3,A3),B2))
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% case_prodI
tff(fact_2956_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,bool))] :
      ( ! [A6: A,B5: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
         => pp(aa(B,bool,aa(A,fun(B,bool),C2,A6),B5)) )
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C2),P3)) ) ).

% case_prodI2
tff(fact_2957_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),A3: B,B2: C] :
      ( pp(aa(set(A),bool,member(A,Z2),aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),B2)))
     => pp(aa(set(A),bool,member(A,Z2),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)))) ) ).

% mem_case_prodI
tff(fact_2958_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),Z2: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A6: A,B5: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
         => pp(aa(set(C),bool,member(C,Z2),aa(B,set(C),aa(A,fun(B,set(C)),C2,A6),B5))) )
     => pp(aa(set(C),bool,member(C,Z2),aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P3))) ) ).

% mem_case_prodI2
tff(fact_2959_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,fun(C,bool))),X: C] :
      ( ! [A6: A,B5: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) = P3 )
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A6),B5),X)) )
     => pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C2),P3),X)) ) ).

% case_prodI2'
tff(fact_2960_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] : bit_take_bit_num(aa(nat,nat,suc,N2),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_2961_take__bit__num__simps_I5_J,axiom,
    ! [R3: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_2962_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N2),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_jd(num,option(num)),bit_take_bit_num(N2,M)) ).

% take_bit_num_simps(3)
tff(fact_2963_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_2964_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N2)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N2)) ) ).

% take_bit_numeral_numeral
tff(fact_2965_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),P3: product_prod(B,C)] :
      ( pp(aa(set(A),bool,member(A,Z2),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),P3)))
     => ~ ! [X3: B,Y3: C] :
            ( ( P3 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y3) )
           => ~ pp(aa(set(A),bool,member(A,Z2),aa(C,set(A),aa(B,fun(C,set(A)),C2,X3),Y3))) ) ) ).

% mem_case_prodE
tff(fact_2966_case__prodD,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),F3,A3),B2)) ) ).

% case_prodD
tff(fact_2967_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,bool)),P3: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C2),P3))
     => ~ ! [X3: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),C2,X3),Y3)) ) ) ).

% case_prodE
tff(fact_2968_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(A,fun(B,fun(C,bool))),A3: A,B2: B,C2: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C2))
     => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R2,A3),B2),C2)) ) ).

% case_prodD'
tff(fact_2969_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,bool))),P3: product_prod(A,B),Z2: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C2),P3),Z2))
     => ~ ! [X3: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,X3),Y3),Z2)) ) ) ).

% case_prodE'
tff(fact_2970_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),aTP_Lamp_je(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_2971_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,fun(B,bool)),B4: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A5),B4))
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A5))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),B4)))) ) ).

% Collect_case_prod_mono
tff(fact_2972_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),N2: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jf(nat,fun(nat,fun(nat,bool)),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jh(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).

% sum.triangle_reindex_eq
tff(fact_2973_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),N2: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jf(nat,fun(nat,fun(nat,bool)),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jj(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).

% prod.triangle_reindex_eq
tff(fact_2974_execute__bind__case,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(B),G3: fun(B,heap_Time_Heap(A)),H: 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_bind(B,A,F3,G3)),H) = case_option(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),none(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),product_case_prod(B,product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_jl(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),G3)),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F3),H)) ).

% execute_bind_case
tff(fact_2975_map__to__set__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_to_set(A,B,M) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_jm(fun(A,option(B)),fun(A,fun(B,bool)),M))) ).

% map_to_set_def
tff(fact_2976_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N2: num,Q3: num] :
          ( ( bit_take_bit_num(M,N2) = aa(num,option(num),some(num),Q3) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N2)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_2977_Heap__Time__Monad_Obind__def,axiom,
    ! [B: $tType,A: $tType,F3: heap_Time_Heap(A),G3: fun(A,heap_Time_Heap(B))] : heap_Time_bind(A,B,F3,G3) = heap_Time_Heap2(B,aa(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_jp(heap_Time_Heap(A),fun(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),F3),G3)) ).

% Heap_Time_Monad.bind_def
tff(fact_2978_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_jq(set(A),fun(set(A),bool)))) ).

% finite_psubset_def
tff(fact_2979_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),bool)] : rel_of(A,B,M,P) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_jr(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),M),P))) ).

% rel_of_def
tff(fact_2980_divmod__int__def,axiom,
    ! [M: num,N2: num] : unique8689654367752047608divmod(int,M,N2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N2))) ).

% divmod_int_def
tff(fact_2981_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N2: num] : unique8689654367752047608divmod(A,M,N2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N2))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2))) ) ).

% divmod_def
tff(fact_2982_divmod_H__nat__def,axiom,
    ! [M: num,N2: num] : unique8689654367752047608divmod(nat,M,N2) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N2))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N2))) ).

% divmod'_nat_def
tff(fact_2983_mlex__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F3,R2) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_js(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F3),R2))) ).

% mlex_eq
tff(fact_2984_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N2: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,N2)) = 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_jt(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N2)) ) ).

% divmod_algorithm_code(6)
tff(fact_2985_int__ge__less__than2__def,axiom,
    ! [D3: int] : int_ge_less_than2(D3) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_ju(int,fun(int,fun(int,bool)),D3))) ).

% int_ge_less_than2_def
tff(fact_2986_int__ge__less__than__def,axiom,
    ! [D3: int] : int_ge_less_than(D3) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_jv(int,fun(int,fun(int,bool)),D3))) ).

% int_ge_less_than_def
tff(fact_2987_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N2: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N2)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N2),unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,aa(num,num,bit1,N2)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_2988_split__part,axiom,
    ! [B: $tType,A: $tType,P: bool,Q: fun(A,fun(B,bool)),X5: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_jw(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q)),X5))
    <=> ( pp(P)
        & pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q),X5)) ) ) ).

% split_part
tff(fact_2989_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2)) ) ).

% semiring_norm(80)
tff(fact_2990_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N2)) ) ).

% semiring_norm(73)
tff(fact_2991_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit1,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N2)) ) ).

% semiring_norm(72)
tff(fact_2992_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit0,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2)) ) ).

% semiring_norm(81)
tff(fact_2993_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),one2)) ).

% semiring_norm(70)
tff(fact_2994_semiring__norm_I77_J,axiom,
    ! [N2: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit1,N2))) ).

% semiring_norm(77)
tff(fact_2995_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit0,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N2)) ) ).

% semiring_norm(74)
tff(fact_2996_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit1,N2)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N2)) ) ).

% semiring_norm(79)
tff(fact_2997_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N2),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(N2,M))) ).

% take_bit_num_simps(4)
tff(fact_2998_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N2: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_2999_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N2: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N2))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N2))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N2)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N2),unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,aa(num,num,bit1,N2)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_3000_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_jx(A,fun(B,bool))),Prod)) ).

% prod.disc_eq_case
tff(fact_3001_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N5))
       => ( ! [N5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N5))
         => ( ! [M5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2)
           => ( ! [M5: num,N5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N5))
             => ( ! [M5: num,N5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N5))
               => ( ! [M5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2)
                 => ( ! [M5: num,N5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N5))
                   => ~ ! [M5: num,N5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N5)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_3002_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N2: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N2)),aa(num,A,numeral_numeral(A),N2))),one_one(A)) ) ).

% numeral_code(3)
tff(fact_3003_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z2: A,W2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))) ) ).

% power_numeral_odd
tff(fact_3004_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),A3) ) ).

% power3_eq_cube
tff(fact_3005_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N2: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N2),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_3006_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),aa(num,num,bit0,one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_3007_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_jd(num,option(num)),bit_take_bit_num(pred_numeral(R3),M)) ).

% take_bit_num_simps(6)
tff(fact_3008_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_3009_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),N2: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jy(nat,fun(nat,fun(nat,bool)),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jj(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% prod.triangle_reindex
tff(fact_3010_take__bit__num__def,axiom,
    ! [N2: nat,M: num] :
      ( ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat) )
       => ( bit_take_bit_num(N2,M) = none(num) ) )
      & ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),aa(num,nat,numeral_numeral(nat),M)) != zero_zero(nat) )
       => ( bit_take_bit_num(N2,M) = aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N2),aa(num,nat,numeral_numeral(nat),M)))) ) ) ) ).

% take_bit_num_def
tff(fact_3011_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_lessThan(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),K)) ) ) ).

% lessThan_iff
tff(fact_3012_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),X)),aa(A,set(A),set_ord_lessThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% lessThan_subset_iff
tff(fact_3013_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3014_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),pred_numeral(K))) ) ).

% less_Suc_numeral
tff(fact_3015_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),pred_numeral(K)),N2)) ) ).

% less_numeral_Suc
tff(fact_3016_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N2))),aa(nat,A,G3,N2)) ) ).

% prod.lessThan_Suc
tff(fact_3017_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),pred_numeral(K))) ) ).

% le_Suc_numeral
tff(fact_3018_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),pred_numeral(K)),N2)) ) ).

% le_numeral_Suc
tff(fact_3019_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R3),M))) ).

% take_bit_num_simps(7)
tff(fact_3020_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] : aa(A,set(A),set_ord_lessThan(A),X) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_3021_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_jz(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_3022_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),N2) = bot_bot(set(nat)) )
    <=> ( N2 = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3023_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [N2: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),N2) = bot_bot(set(A)) )
        <=> ( N2 = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_3024_finite__nat__iff__bounded,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_lessThan(nat),K3))) ) ).

% finite_nat_iff_bounded
tff(fact_3025_finite__nat__bounded,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => ? [K2: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_lessThan(nat),K2))) ) ).

% finite_nat_bounded
tff(fact_3026_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M)),aa(A,set(A),set_ord_lessThan(A),N2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N2)) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3027_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ka(fun(nat,A),fun(nat,fun(nat,A)),G3),N2)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% sum.nat_diff_reindex
tff(fact_3028_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_kb(fun(nat,A),fun(nat,fun(nat,A)),G3),N2)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% prod.nat_diff_reindex
tff(fact_3029_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_3030_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_3031_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3032_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_3033_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N2: A] :
          ( ! [X3: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P,X3)))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),aa(A,set(A),set_ord_lessThan(A),N2))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),aa(A,set(A),set_ord_lessThan(A),N2))) = 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_kc(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),N2)) ) ) ) ).

% sum_diff_distrib
tff(fact_3034_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3035_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ey(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3036_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dg(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M)),aa(nat,A,F3,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3037_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_er(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3038_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N2)) = N2 ) ) ).

% numeral_num_of_nat
tff(fact_3039_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3040_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),one_one(nat)))
     => ( num_of_nat(N2) = one2 ) ) ).

% num_of_nat_One
tff(fact_3041_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_er(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3042_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3043_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_kd(fun(nat,A),fun(nat,fun(nat,A)),G3),K)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))) ) ).

% sum.nat_group
tff(fact_3044_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ke(fun(nat,A),fun(nat,fun(nat,A)),G3),K)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))) ) ).

% prod.nat_group
tff(fact_3045_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_kf(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hl(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N2)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% sum.nested_swap'
tff(fact_3046_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_kg(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N2)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% prod.nested_swap'
tff(fact_3047_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_3048_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_3049_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% one_diff_power_eq
tff(fact_3050_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% power_diff_1_eq
tff(fact_3051_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N2: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3052_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_3053_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dd(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% sum.atMost_shift
tff(fact_3054_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_er(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% prod.atMost_shift
tff(fact_3055_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)) = aa(num,num,bit0,num_of_nat(N2)) ) ) ).

% num_of_nat_double
tff(fact_3056_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(N2)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_3057_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(nat,A,semiring_1_of_nat(A),N2) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_strict
tff(fact_3058_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_kh(A,fun(nat,fun(A,fun(nat,A))),X),N2),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N2)))) ) ).

% diff_power_eq_sum
tff(fact_3059_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ki(A,fun(nat,fun(A,fun(nat,A))),X),N2),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% power_diff_sumr2
tff(fact_3060_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_numeral_bit0
tff(fact_3061_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N2: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_kj(A,fun(nat,fun(nat,A)),X),N2)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).

% one_diff_power_eq'
tff(fact_3062_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),N2: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jy(nat,fun(nat,fun(nat,bool)),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jh(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) ) ).

% sum.triangle_reindex
tff(fact_3063_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N2)))) ) ).

% mask_numeral
tff(fact_3064_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list(bool)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(list(bool),int,aa(int,fun(list(bool),int),aa(fun(bool,int),fun(int,fun(list(bool),int)),groups4207007520872428315er_sum(bool,int),zero_neq_one_of_bool(int)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(list(bool),nat,size_size(list(bool)),Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_3065_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_3066_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_3067_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_kk(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_3068_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).

% mask_nat_positive_iff
tff(fact_3069_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344971392196577ns_xor(int,K,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_3070_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344971392196577ns_xor(int,K,L)),zero_zero(int)))
    <=> ~ ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% xor_negative_int_iff
tff(fact_3071_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_3072_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_3073_divmod__integer_H__def,axiom,
    ! [M: num,N2: num] : unique8689654367752047608divmod(code_integer,M,N2) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),N2))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),N2))) ).

% divmod_integer'_def
tff(fact_3074_exhaustive__integer_H_Ocases,axiom,
    ! [X: product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F2: fun(code_integer,option(product_prod(bool,list(code_term)))),D2: code_integer,I3: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(code_integer,option(product_prod(bool,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),F2),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D2),I3)) ).

% exhaustive_integer'.cases
tff(fact_3075_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F2: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: code_integer,I3: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,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(bool,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),F2),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D2),I3)) ).

% full_exhaustive_integer'.cases
tff(fact_3076_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_kl(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_3077_less__eq__integer__code_I1_J,axiom,
    pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_eq_integer_code(1)
tff(fact_3078_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = zero_zero(code_integer) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = one_one(code_integer) ) ) ) ) ) ).

% sgn_integer_code
tff(fact_3079_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_3080_less__eq__mask,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),bit_se2239418461657761734s_mask(nat,N2))) ).

% less_eq_mask
tff(fact_3081_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344971392196577ns_xor(int,X,Y))) ) ) ).

% XOR_lower
tff(fact_3082_mask__nonnegative__int,axiom,
    ! [N2: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,N2))) ).

% mask_nonnegative_int
tff(fact_3083_not__mask__negative__int,axiom,
    ! [N2: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2239418461657761734s_mask(int,N2)),zero_zero(int))) ).

% not_mask_negative_int
tff(fact_3084_less__mask,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),bit_se2239418461657761734s_mask(nat,N2))) ) ).

% less_mask
tff(fact_3085_mask__nat__less__exp,axiom,
    ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),bit_se2239418461657761734s_mask(nat,N2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ).

% mask_nat_less_exp
tff(fact_3086_finite__int__iff__bounded,axiom,
    ! [S: set(int)] :
      ( pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S)),aa(int,set(int),set_ord_lessThan(int),K3))) ) ).

% finite_int_iff_bounded
tff(fact_3087_XOR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344971392196577ns_xor(int,X,Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ) ) ) ).

% XOR_upper
tff(fact_3088_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_km(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3089_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( aa(int,code_integer,code_integer_of_int,K) = aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( ( ( K = zero_zero(int) )
           => ( aa(int,code_integer,code_integer_of_int,K) = zero_zero(code_integer) ) )
          & ( ( K != zero_zero(int) )
           => ( aa(int,code_integer,code_integer_of_int,K) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_3090_integer__of__num_I3_J,axiom,
    ! [N2: num] : code_integer_of_num(aa(num,num,bit1,N2)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_num(N2)),code_integer_of_num(N2))),one_one(code_integer)) ).

% integer_of_num(3)
tff(fact_3091_take__bit__num__code,axiom,
    ! [N2: nat,M: num] : bit_take_bit_num(N2,M) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_kq(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N2),M)) ).

% take_bit_num_code
tff(fact_3092_even__sum__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(B),nat,finite_card(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),A5),F3))))) ) ) ) ).

% even_sum_iff
tff(fact_3093_card__Collect__less__nat,axiom,
    ! [N2: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2))) = N2 ).

% card_Collect_less_nat
tff(fact_3094_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,L,U))) ).

% finite_atLeastLessThan_integer
tff(fact_3095_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or1337092689740270186AtMost(code_integer,L,U))) ).

% finite_atLeastAtMost_integer
tff(fact_3096_card__Collect__le__nat,axiom,
    ! [N2: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ca(nat,fun(nat,bool)),N2))) = aa(nat,nat,suc,N2) ).

% card_Collect_le_nat
tff(fact_3097_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_3098_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ks(A,fun(B,A),Y)),A5) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(set(B),nat,finite_card(B),A5)) ) ).

% prod_constant
tff(fact_3099_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,A),V2: num] : case_nat(A,A3,F3,aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,F3,pred_numeral(V2)) ).

% case_nat_numeral
tff(fact_3100_card__0__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) )
      <=> ( A5 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_3101_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_kt(A,fun(B,A),Y)),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),Y) ) ).

% sum_constant
tff(fact_3102_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,A),V2: num,N2: nat] : case_nat(A,A3,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N2)) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N2)) ).

% case_nat_add_eq_if
tff(fact_3103_sum__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A5: set(B),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),A5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ku(fun(B,bool),fun(B,A),P)),A5) = aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(fun(B,bool),set(B),collect(B),P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_3104_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),K) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = K ) ) ) ).

% abs_integer_code
tff(fact_3105_less__integer__code_I1_J,axiom,
    ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_integer_code(1)
tff(fact_3106_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Xa2),X)) ) ).

% less_integer.abs_eq
tff(fact_3107_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U))) ).

% finite_atLeastZeroLessThan_integer
tff(fact_3108_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(nat,A),Nat: nat] : aa(A,B,H,case_nat(A,F1,F22,Nat)) = case_nat(B,aa(A,B,H,F1),aa(fun(nat,A),fun(nat,B),aTP_Lamp_kv(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_3109_num_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(num,A),F32: fun(num,A),Num: num] : aa(A,B,H,case_num(A,F1,F22,F32,Num)) = case_num(B,aa(A,B,H,F1),aa(fun(num,A),fun(num,B),aTP_Lamp_kw(fun(A,B),fun(fun(num,A),fun(num,B)),H),F22),aa(fun(num,A),fun(num,B),aTP_Lamp_kw(fun(A,B),fun(fun(num,A),fun(num,B)),H),F32),Num) ).

% num.case_distrib
tff(fact_3110_n__subsets,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(nat,fun(set(A),bool),aTP_Lamp_kx(set(A),fun(nat,fun(set(A),bool)),A5),K))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A5)),K) ) ) ).

% n_subsets
tff(fact_3111_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A)] : case_nat(A,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_3112_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X2: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ).

% old.nat.simps(5)
tff(fact_3113_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A5: set(A),N2: nat] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ? [B10: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B10))
          & ( aa(set(A),nat,finite_card(A),B10) = N2 )
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B10),A5)) ) ) ).

% infinite_arbitrarily_large
tff(fact_3114_card__subset__eq,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
       => ( ( aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,finite_card(A),B4) )
         => ( A5 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3115_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B4: set(A),A5: set(B),R3: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( ! [A6: B] :
            ( pp(aa(set(B),bool,member(B,A6),A5))
           => ? [B11: A] :
                ( pp(aa(set(A),bool,member(A,B11),B4))
                & pp(aa(A,bool,aa(B,fun(A,bool),R3,A6),B11)) ) )
       => ( ! [A12: B,A23: B,B5: A] :
              ( pp(aa(set(B),bool,member(B,A12),A5))
             => ( pp(aa(set(B),bool,member(B,A23),A5))
               => ( pp(aa(set(A),bool,member(A,B5),B4))
                 => ( pp(aa(A,bool,aa(B,fun(A,bool),R3,A12),B5))
                   => ( pp(aa(A,bool,aa(B,fun(A,bool),R3,A23),B5))
                     => ( A12 = A23 ) ) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A5)),aa(set(A),nat,finite_card(A),B4))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3116_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Xa2),X)) ) ).

% less_eq_integer.abs_eq
tff(fact_3117_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T6: set(B),R2: fun(A,fun(B,bool)),K: fun(B,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => ( pp(aa(set(B),bool,finite_finite2(B),T6))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),T6))
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ig(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S2),R2),X3))) = aa(B,nat,K,X3) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_kz(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T6),R2)),S2) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_3118_card__eq__sum,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_la(A,nat)),A5) ).

% card_eq_sum
tff(fact_3119_card__eq__0__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) )
    <=> ( ( A5 = bot_bot(set(A)) )
        | ~ pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% card_eq_0_iff
tff(fact_3120_card__ge__0__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5)))
     => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ).

% card_ge_0_finite
tff(fact_3121_card__image__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(A),nat,finite_card(A),A5))) ) ).

% card_image_le
tff(fact_3122_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C4: nat] :
      ( ! [G5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),G5),F4))
         => ( pp(aa(set(A),bool,finite_finite2(A),G5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G5)),C4)) ) )
     => ( pp(aa(set(A),bool,finite_finite2(A),F4))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C4)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3123_card__seteq,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B4)),aa(set(A),nat,finite_card(A),A5)))
         => ( A5 = B4 ) ) ) ) ).

% card_seteq
tff(fact_3124_card__mono,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4))) ) ) ).

% card_mono
tff(fact_3125_obtain__subset__with__card__n,axiom,
    ! [A: $tType,N2: nat,S: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(set(A),nat,finite_card(A),S)))
     => ~ ! [T7: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T7),S))
           => ( ( aa(set(A),nat,finite_card(A),T7) = N2 )
             => ~ pp(aa(set(A),bool,finite_finite2(A),T7)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3126_card__less__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3127_card__le__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3128_psubset__card__mono,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4))) ) ) ).

% psubset_card_mono
tff(fact_3129_card__Un__le,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)))) ).

% card_Un_le
tff(fact_3130_card__less__Suc2,axiom,
    ! [M6: set(nat),I2: nat] :
      ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_lb(set(nat),fun(nat,fun(nat,bool)),M6),I2))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_lc(set(nat),fun(nat,fun(nat,bool)),M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_3131_card__less__Suc,axiom,
    ! [M6: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_lb(set(nat),fun(nat,fun(nat,bool)),M6),I2)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_lc(set(nat),fun(nat,fun(nat,bool)),M6),I2))) ) ) ).

% card_less_Suc
tff(fact_3132_card__less,axiom,
    ! [M6: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_lc(set(nat),fun(nat,fun(nat,bool)),M6),I2))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3133_subset__card__intvl__is__intvl,axiom,
    ! [A5: set(nat),K: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),A5),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A5)))))
     => ( A5 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A5))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_3134_sum__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ld(fun(A,nat),fun(A,nat),F3)),A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,finite_card(A),A5)) ).

% sum_Suc
tff(fact_3135_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S: set(A),T2: set(B),R2: fun(A,fun(B,bool)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(B),bool,finite_finite2(B),T2))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),T2))
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ig(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S),R2),X3))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_kz(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T2),R2)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T2)) ) ) ) ) ).

% sum_multicount
tff(fact_3136_card__gt__0__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5)))
    <=> ( ( A5 != bot_bot(set(A)) )
        & pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% card_gt_0_iff
tff(fact_3137_sum__bounded__below,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A5: set(B),K5: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K5),aa(B,A,F3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5))) ) ) ).

% sum_bounded_below
tff(fact_3138_sum__bounded__above,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A5: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),K5)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),K5))) ) ) ).

% sum_bounded_above
tff(fact_3139_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),image2(A,B,F3),A5)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B4)),aa(set(A),nat,finite_card(A),A5))) ) ) ).

% surj_card_le
tff(fact_3140_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(nat,nat,suc,zero_zero(nat))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => ! [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),A5))
               => ( X4 = Xa ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3141_card__Diff__subset,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ).

% card_Diff_subset
tff(fact_3142_card__psubset,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4)) ) ) ) ).

% card_psubset
tff(fact_3143_card__Un__Int,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ) ).

% card_Un_Int
tff(fact_3144_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)))) ) ).

% diff_card_le_card_Diff
tff(fact_3145_card__Diff__subset__Int,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ).

% card_Diff_subset_Int
tff(fact_3146_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set(nat),N2: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N7),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N7)),N2)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_3147_card__sum__le__nat__sum,axiom,
    ! [S: set(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ez(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ez(nat,nat)),S))) ).

% card_sum_le_nat_sum
tff(fact_3148_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A5)))
     => ( A5 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_3149_card__Un__disjoint,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_3150_integer__of__num_I2_J,axiom,
    ! [N2: num] : code_integer_of_num(aa(num,num,bit0,N2)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_num(N2)),code_integer_of_num(N2)) ).

% integer_of_num(2)
tff(fact_3151_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F3: fun(B,A),N2: A,K: nat] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),N2)) ) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A5)),K))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N2),K))) ) ) ) ) ).

% prod_le_power
tff(fact_3152_sum__bounded__above__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_field(A)
     => ! [A5: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),K5),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))))) )
         => ( pp(aa(set(B),bool,finite_finite2(B),A5))
           => ( ( A5 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),K5)) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3153_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A5: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I3)),K5)) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A5)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),K5))) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3154_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S: set(A),R2: set(B),G3: fun(A,B),F3: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(set(B),bool,finite_finite2(B),R2))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G3),S)),R2))
             => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_le(fun(A,B),fun(fun(B,C),fun(A,C)),G3),F3)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_lg(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G3),F3)),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_3155_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_lh(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),S)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_lh(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,finite_card(B),S)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_3156_card__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),N2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_li(set(A),fun(nat,fun(list(A),bool)),A5),N2))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A5))),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ) ).

% card_lists_length_le
tff(fact_3157_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
       => ( code_num_of_integer(K) = one2 ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
       => ( code_num_of_integer(K) = aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_lj(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% num_of_integer_code
tff(fact_3158_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,bool,fNot,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),K))) ).

% bit_cut_integer_def
tff(fact_3159_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,int,code_int_of_integer,K) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( ( ( K = zero_zero(code_integer) )
           => ( aa(code_integer,int,code_int_of_integer,K) = zero_zero(int) ) )
          & ( ( K != zero_zero(code_integer) )
           => ( aa(code_integer,int,code_int_of_integer,K) = aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_lk(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_3160_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L)),modulo_modulo(code_integer,K,L)) ).

% divmod_integer_def
tff(fact_3161_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_3162_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B4))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,member(A,X4),B4)) ) ) ).

% subset_code(1)
tff(fact_3163_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys2)
           => ( ( aa(list(A),set(A),set2(A),Ys2) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys2 = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_3164_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_ll(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_3165_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_lm(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_3166_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Y)))
    <=> pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Y)) ) ).

% int_of_integer_less_iff
tff(fact_3167_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L))) ) ).

% integer_less_iff
tff(fact_3168_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2))) ) ).

% less_integer.rep_eq
tff(fact_3169_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L))) ) ).

% integer_less_eq_iff
tff(fact_3170_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),X),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2))) ) ).

% less_eq_integer.rep_eq
tff(fact_3171_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_pos_if_in_set
tff(fact_3172_nth__mem,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,member(A,aa(nat,A,nth(A,Xs),N2)),aa(list(A),set(A),set2(A),Xs))) ) ).

% nth_mem
tff(fact_3173_list__ball__nth,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X3)) )
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N2))) ) ) ).

% list_ball_nth
tff(fact_3174_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3175_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),X: A] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(A,bool,P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3176_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
    <=> ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_3177_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),L)))
         => pp(aa(A,bool,P,X4)) )
    <=> ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),L)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,L),I4))) ) ) ).

% all_set_conv_nth
tff(fact_3178_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% card_length
tff(fact_3179_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N2))
    <=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% less_eq_nat.simps(2)
tff(fact_3180_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: B,L: list(B)] :
      ( pp(aa(set(A),bool,member(A,aa(B,A,F3,X)),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),L))))
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(B),nat,size_size(list(B)),L)))
          & ( aa(B,A,F3,aa(nat,B,nth(B,L),I4)) = aa(B,A,F3,X) ) ) ) ).

% in_set_image_conv_nth
tff(fact_3181_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L3: list(A),F3: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L3) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L)))
           => ( aa(A,B,F3,aa(nat,A,nth(A,L),I3)) = aa(A,B,F3,aa(nat,A,nth(A,L3),I3)) ) )
       => ( aa(set(A),set(B),image2(A,B,F3),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image2(A,B,F3),aa(list(A),set(A),set2(A),L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_3182_finite__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),N2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_ln(set(A),fun(nat,fun(list(A),bool)),A5),N2)))) ) ).

% finite_lists_length_eq
tff(fact_3183_diff__Suc,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N2)) = case_nat(nat,zero_zero(nat),aTP_Lamp_ez(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)) ).

% diff_Suc
tff(fact_3184_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ~ ! [L4: list(A)] :
                ( sorted_wrt(A,ord_less(A),L4)
               => ( ( aa(list(A),set(A),set2(A),L4) = A5 )
                 => ( aa(list(A),nat,size_size(list(A)),L4) != aa(set(A),nat,finite_card(A),A5) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_3185_card__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),N2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_ln(set(A),fun(nat,fun(list(A),bool)),A5),N2))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A5)),N2) ) ) ).

% card_lists_length_eq
tff(fact_3186_finite__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),N2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_li(set(A),fun(nat,fun(list(A),bool)),A5),N2)))) ) ).

% finite_lists_length_le
tff(fact_3187_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),zero_zero(code_integer)),fFalse) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,bool)),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool)),aTP_Lamp_lo(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_3188_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( code_nat_of_integer(K) = zero_zero(nat) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( code_nat_of_integer(K) = aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_lp(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% nat_of_integer_code
tff(fact_3189_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),set(A),set2(A),union(A,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ).

% set_union
tff(fact_3190_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A5)))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_lq(set(A),fun(nat,fun(list(A),bool)),A5),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ez(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_3191_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A5: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(set(A),nat,finite_card(A),A5)))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),bool),aTP_Lamp_lr(nat,fun(set(A),fun(list(A),bool)),K),A5))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ez(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_3192_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
     => ( code_nat_of_integer(K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_3193_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),N2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_lq(set(A),fun(nat,fun(list(A),bool)),A5),N2)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_3194_distinct__finite__set,axiom,
    ! [A: $tType,X: set(A)] : pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_ls(set(A),fun(list(A),bool),X)))) ).

% distinct_finite_set
tff(fact_3195_strict__sorted__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less(A),L)
        <=> ( sorted_wrt(A,ord_less_eq(A),L)
            & distinct(A,L) ) ) ) ).

% strict_sorted_iff
tff(fact_3196_distinct__length__le,axiom,
    ! [A: $tType,Ys2: list(A),Xs: list(A)] :
      ( distinct(A,Ys2)
     => ( ( aa(list(A),set(A),set2(A),Ys2) = aa(list(A),set(A),set2(A),Xs) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% distinct_length_le
tff(fact_3197_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys2)
             => ( distinct(A,Ys2)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys2) )
                 => ( Xs = Ys2 ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_3198_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,J: nat] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I2 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_3199_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
         => ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xs),I4) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_3200_finite__set__image,axiom,
    ! [A: $tType,A5: set(list(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A5)))
     => ( ! [Xs2: list(A)] :
            ( pp(aa(set(list(A)),bool,member(list(A),Xs2),A5))
           => distinct(A,Xs2) )
       => pp(aa(set(list(A)),bool,finite_finite2(list(A)),A5)) ) ) ).

% finite_set_image
tff(fact_3201_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_3202_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A5 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y4: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y4) = A5 )
                    & sorted_wrt(A,ord_less_eq(A),Y4)
                    & distinct(A,Y4) )
                 => ( Y4 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_3203_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ? [X3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs)))
            & ( aa(nat,A,nth(A,Xs),X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y4),aa(list(A),nat,size_size(list(A)),Xs)))
                  & ( aa(nat,A,nth(A,Xs),Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_3204_distinct__finite__subset,axiom,
    ! [A: $tType,X: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),X))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_lt(set(A),fun(list(A),bool),X)))) ) ).

% distinct_finite_subset
tff(fact_3205_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).

% divmod_abs_code(5)
tff(fact_3206_distinct__sorted__strict__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I2: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J)))
                <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J)) ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_3207_distinct__sorted__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I2: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( distinct(A,L)
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J))) ) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_3208_distinct__sorted__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I2: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J)))
                <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J)) ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_3209_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),X))
     => ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),Y))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),code_nat_of_integer(X)),code_nat_of_integer(Y)))
        <=> pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Y)) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_3210_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),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_3211_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).

% divmod_abs_def
tff(fact_3212_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_lu(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( ( L = zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
              & ( ( L != zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = 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)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_lv(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_3213_mergesort__remdups__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( 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_3214_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_3215_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_lw(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_3216_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_3217_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),X: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),aa(C,B,F3,Y)) ).

% apsnd_conv
tff(fact_3218_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys2: list(A)] :
      ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
     => ( aa(list(A),set(A),set2(A),Zs) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ) ) ).

% set_shuffles
tff(fact_3219_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).

% count_le_length
tff(fact_3220_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
         => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_3221_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_lx(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_3222_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
    <=> ( ( ( Nat = zero_zero(nat) )
         => pp(aa(A,bool,P,F1)) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(1)
tff(fact_3223_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ pp(aa(A,bool,P,F1)) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(2)
tff(fact_3224_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list(bool),N2: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Bs)),N2))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(bool),nat,size_size(list(bool)),Bs)))
            & pp(aa(nat,bool,nth(bool,Bs),N2)) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3225_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,fun(A,A)),V2: num,N2: nat] : aa(nat,A,rec_nat(A,A3,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N2)) = aa(A,A,aa(nat,fun(A,A),F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N2)),aa(nat,A,rec_nat(A,A3,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N2))) ).

% rec_nat_add_eq_if
tff(fact_3226_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,bool)) ) ) ).

% bit_0_eq
tff(fact_3227_old_Onat_Osimps_I6_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T))] : aa(nat,T,rec_nat(T,F1,F22),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_3228_old_Onat_Osimps_I7_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T)),Nat: nat] : aa(nat,T,rec_nat(T,F1,F22),aa(nat,nat,suc,Nat)) = aa(T,T,aa(nat,fun(T,T),F22,Nat),aa(nat,T,rec_nat(T,F1,F22),Nat)) ).

% old.nat.simps(7)
tff(fact_3229_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_3230_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N2),K)),zero_zero(int)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ).

% signed_take_bit_negative_iff
tff(fact_3231_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,fun(A,A)),V2: num] : aa(nat,A,rec_nat(A,A3,F3),aa(num,nat,numeral_numeral(nat),V2)) = aa(A,A,aa(nat,fun(A,A),F3,pred_numeral(V2)),aa(nat,A,rec_nat(A,A3,F3),pred_numeral(V2))) ).

% rec_nat_numeral
tff(fact_3232_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K)),N2))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ) ).

% bit_nat_iff
tff(fact_3233_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N2: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3)),N2))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N2)) ) ) ) ).

% bit_take_bit_iff
tff(fact_3234_rec__nat__0__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_3235_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N2: nat] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,aa(nat,nat,suc,N2)) = aa(A,A,aa(nat,fun(A,A),F22,N2),aa(nat,A,F3,N2)) ) ) ).

% rec_nat_Suc_imp
tff(fact_3236_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,N2: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W2))),N2))
        <=> pp(case_nat(bool,fFalse,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),N2)) ) ) ).

% bit_numeral_rec(1)
tff(fact_3237_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,N2: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W2))),N2))
        <=> pp(case_nat(bool,fTrue,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),N2)) ) ) ).

% bit_numeral_rec(2)
tff(fact_3238_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3239_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N5: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),M4))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),M4))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N5)) ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N5),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N5)) ) ) ) ).

% int_bit_bound
tff(fact_3240_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,N2: nat] :
          ( ! [J2: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J2)))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),N2))
          <=> ( ( ( N2 = zero_zero(nat) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)) )
              & ( ( N2 != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N2)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3241_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_ez(nat,nat),Nat) ).

% pred_def
tff(fact_3242_ranI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M,A3) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(A),bool,member(A,B2),ran(B,A,M))) ) ).

% ranI
tff(fact_3243_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N2: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N2),A3) = if(A,aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2)),A3)),N2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2)),A3)),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N2),aa(A,A,uminus_uminus(A),one_one(A)))),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2)),A3)) ) ).

% signed_take_bit_code
tff(fact_3244_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),L: list(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A5 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A5) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_3245_finite__enumerate,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => ? [R: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(nat),nat,finite_card(nat),S)))
          & ! [N8: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N8),aa(set(nat),nat,finite_card(nat),S)))
             => pp(aa(set(nat),bool,member(nat,aa(nat,nat,R,N8)),S)) ) ) ) ).

% finite_enumerate
tff(fact_3246_card__partition,axiom,
    ! [A: $tType,C4: set(set(A)),K: nat] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),C4))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)))
       => ( ! [C3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),C3),C4))
             => ( aa(set(A),nat,finite_card(A),C3) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),C1),C4))
               => ( pp(aa(set(set(A)),bool,member(set(A),C22),C4))
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C4)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)) ) ) ) ) ) ).

% card_partition
tff(fact_3247_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_3248_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se1065995026697491101ons_or(int,K,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_3249_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se4730199178511100633sh_bit(int,N2,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_3250_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se1065995026697491101ons_or(int,K,L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% or_negative_int_iff
tff(fact_3251_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se4730199178511100633sh_bit(int,N2,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% push_bit_negative_int_iff
tff(fact_3252_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_3253_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_3254_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_3255_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,X)) = X ) ) ) ).

% cSup_atLeastLessThan
tff(fact_3256_cSUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(B),C2: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ly(A,fun(B,A),C2)),A5)) = C2 ) ) ) ).

% cSUP_const
tff(fact_3257_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [A6: A] :
            ( pp(aa(set(A),bool,member(A,A6),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,A6))) )
       => pp(aa(set(B),bool,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5)))) ) ) ).

% finite_UN_I
tff(fact_3258_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_3259_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N2),A3) = bit_se4730199178511100633sh_bit(A,N2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% push_bit_Suc
tff(fact_3260_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_3261_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_3262_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z2: A,X6: set(A)] :
          ( pp(aa(set(A),bool,member(A,Z2),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = Z2 ) ) ) ) ).

% cSup_eq_maximum
tff(fact_3263_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
         => ( ! [Y3: A] :
                ( ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ).

% cSup_eq
tff(fact_3264_Some__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A5)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A5)) ) ) ) ).

% Some_Sup
tff(fact_3265_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X6)),Z2)) ) ) ) ).

% cSup_least
tff(fact_3266_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
           => ( ! [Y3: A] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),Y3)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3267_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% le_cSup_finite
tff(fact_3268_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
         => ( ( X6 != bot_bot(set(A)) )
           => ~ ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),X6))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X3)) ) ) ) ) ).

% less_cSupE
tff(fact_3269_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X6)))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X3)) ) ) ) ) ).

% less_cSupD
tff(fact_3270_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),A3)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3)) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_3271_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se1065995026697491101ons_or(int,X,Y))) ) ) ).

% OR_lower
tff(fact_3272_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),bit_se1065995026697491101ons_or(int,K,L))) ) ).

% or_greater_eq
tff(fact_3273_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_se4730199178511100633sh_bit(int,M,K)),N2))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_3274_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,M,Q3)),N2))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3275_finite__UNION__then__finite,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B),A3: B] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))))
     => ( pp(aa(set(B),bool,member(B,A3),A5))
       => pp(aa(set(A),bool,finite_finite2(A),aa(B,set(A),B4,A3))) ) ) ).

% finite_UNION_then_finite
tff(fact_3276_Some__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ( A5 != 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,F3),A5))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_lz(fun(A,B),fun(A,option(B)),F3)),A5)) ) ) ) ).

% Some_SUP
tff(fact_3277_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2))) ).

% card_Union_le_sum_card
tff(fact_3278_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F3: fun(A,B),A5: set(A),X: A,Y: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( pp(aa(set(A),bool,member(A,Y),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_3279_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A),R3: A,S2: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( pp(aa(set(A),bool,member(A,R3),A5))
           => ( pp(aa(set(A),bool,member(A,S2),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R3),S2))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R3)),aa(A,B,F3,S2))) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_3280_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [R: A,S5: A] :
              ( pp(aa(set(A),bool,member(A,R),A5))
             => ( pp(aa(set(A),bool,member(A,S5),A5))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R),S5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R)),aa(A,B,F3,S5))) ) ) )
         => strict_mono_on(A,B,F3,A5) ) ) ).

% strict_mono_onI
tff(fact_3281_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( strict_mono_on(A,B,F3,A5)
        <=> ! [R4: A,S6: A] :
              ( ( pp(aa(set(A),bool,member(A,R4),A5))
                & pp(aa(set(A),bool,member(A,S6),A5))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R4),S6)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R4)),aa(A,B,F3,S6))) ) ) ) ).

% strict_mono_on_def
tff(fact_3282_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,set(A)),G3: fun(C,set(B)),X: C,X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),aa(C,set(B),G3,X)))),X6))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(C,set(B),G3,X)),aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ma(fun(B,set(A)),fun(set(A),fun(B,bool)),F3),X6)))) ) ).

% UN_image_subset
tff(fact_3283_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_3284_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_3285_cSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(B),F3: fun(B,A),M6: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),M6)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),M6)) ) ) ) ).

% cSUP_least
tff(fact_3286_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_3287_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A5))
                 => ( pp(aa(set(A),bool,member(A,Y3),A5))
                   => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)),A5)) ) )
             => pp(aa(set(A),bool,member(A,aa(set(A),A,complete_Sup_Sup(A),A5)),A5)) ) ) ) ) ).

% finite_Sup_in
tff(fact_3288_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A3)) ) ) ) ).

% cSup_abs_le
tff(fact_3289_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),U2))
         => pp(aa(set(A),bool,finite_finite2(A),X3)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2))) ) ).

% card_Union_le_sum_card_weak
tff(fact_3290_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),bot_bot(set(B))))) = A5 ).

% Union_image_empty
tff(fact_3291_UN__le__add__shift,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_mb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_atMost(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M6),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K)))) ).

% UN_le_add_shift
tff(fact_3292_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_mb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M6),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K)))) ).

% UN_le_add_shift_strict
tff(fact_3293_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S)),L))),E3)) ) ) ) ).

% cSup_asclose
tff(fact_3294_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] : bit_se4730199178511100633sh_bit(A,N2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,N2,A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% push_bit_double
tff(fact_3295_dvd__partition,axiom,
    ! [A: $tType,C4: set(set(A)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),C4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X3))) )
       => ( ! [X3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X3),C4))
             => ! [Xa4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),Xa4),C4))
                 => ( ( X3 != Xa4 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)))) ) ) ) ).

% dvd_partition
tff(fact_3296_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat,A3: A] : bit_se4730199178511100633sh_bit(A,N2,A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N2)) ) ).

% push_bit_eq_mult
tff(fact_3297_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_3298_sum_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),A5: fun(B,set(C)),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A5,X3))) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),I5))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,member(B,Xa4),I5))
                     => ( ( X3 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A5,X3)),aa(B,set(C),A5,Xa4)) = bot_bot(set(C)) ) ) ) )
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A5),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_mc(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A5),G3)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_3299_prod_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),A5: fun(B,set(C)),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A5,X3))) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),I5))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,member(B,Xa4),I5))
                     => ( ( X3 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A5,X3)),aa(B,set(C),A5,Xa4)) = bot_bot(set(C)) ) ) ) )
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A5),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_md(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A5),G3)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_3300_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),I5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_me(fun(A,set(B)),fun(A,nat),A5)),I5))) ) ).

% card_UN_le
tff(fact_3301_UN__le__eq__Un0,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M6),aa(nat,set(nat),set_ord_atMost(nat),N2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M6),set_or1337092689740270186AtMost(nat,one_one(nat),N2)))),aa(nat,set(A),M6,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_3302_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),I5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),I5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A5,X3))) )
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),I5))
             => ! [Xa4: A] :
                  ( pp(aa(set(A),bool,member(A,Xa4),I5))
                 => ( ( X3 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X3)),aa(A,set(B),A5,Xa4)) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_me(fun(A,set(B)),fun(A,nat),A5)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_3303_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N2: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N2)) = bit_se1065995026697491101ons_or(A,one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,N2))) ) ).

% mask_Suc_double
tff(fact_3304_OR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se1065995026697491101ons_or(int,X,Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N2))) ) ) ) ).

% OR_upper
tff(fact_3305_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N2: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_mf(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)) ) ).

% take_bit_sum
tff(fact_3306_UN__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C4: set(C),A5: fun(C,set(D)),B4: set(D)] :
      ( ( ( C4 = bot_bot(set(C)) )
       => ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_mg(fun(C,set(D)),fun(set(D),fun(C,set(D))),A5),B4)),C4)) = bot_bot(set(D)) ) )
      & ( ( C4 != bot_bot(set(C)) )
       => ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_mg(fun(C,set(D)),fun(set(D),fun(C,set(D))),A5),B4)),C4)) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A5),C4))),B4) ) ) ) ).

% UN_simps(2)
tff(fact_3307_UN__simps_I3_J,axiom,
    ! [E: $tType,F: $tType,C4: set(F),A5: set(E),B4: fun(F,set(E))] :
      ( ( ( C4 = bot_bot(set(F)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_mh(set(E),fun(fun(F,set(E)),fun(F,set(E))),A5),B4)),C4)) = bot_bot(set(E)) ) )
      & ( ( C4 != bot_bot(set(F)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_mh(set(E),fun(fun(F,set(E)),fun(F,set(E))),A5),B4)),C4)) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A5),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),B4),C4))) ) ) ) ).

% UN_simps(3)
tff(fact_3308_UN__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A5: set(B),B4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))) = 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),M6),A5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),B4))) ).

% UN_Un
tff(fact_3309_UN__constant,axiom,
    ! [B: $tType,A: $tType,A5: set(B),C2: set(A)] :
      ( ( ( A5 = bot_bot(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_mi(set(A),fun(B,set(A)),C2)),A5)) = bot_bot(set(A)) ) )
      & ( ( A5 != bot_bot(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_mi(set(A),fun(B,set(A)),C2)),A5)) = C2 ) ) ) ).

% UN_constant
tff(fact_3310_SUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F3: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mj(A,fun(B,A),F3)),A5)) = F3 ) ) ) ).

% SUP_const
tff(fact_3311_subset__mset_OcSUP__const,axiom,
    ! [B: $tType,A: $tType,A5: set(B),C2: multiset(A)] :
      ( ( A5 != bot_bot(set(B)) )
     => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aTP_Lamp_mk(multiset(A),fun(B,multiset(A)),C2)),A5)) = C2 ) ) ).

% subset_mset.cSUP_const
tff(fact_3312_Sup__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A5: set(fun(A,B)),X: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A5),X) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_ml(A,fun(fun(A,B),B),X)),A5)) ) ).

% Sup_apply
tff(fact_3313_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A5) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => ( X4 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3314_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A5) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => ( X4 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3315_Union__Un__distrib,axiom,
    ! [A: $tType,A5: set(set(A)),B4: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),A5),B4)) = 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)),A5)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) ).

% Union_Un_distrib
tff(fact_3316_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_3317_SUP__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_mm(A,A)),A5)) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ).

% SUP_identity_eq
tff(fact_3318_SUP__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [F3: fun(C,fun(B,A)),A5: set(C),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),aa(set(C),set(fun(B,A)),image2(C,fun(B,A),F3),A5)),X) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_mn(fun(C,fun(B,A)),fun(B,fun(C,A)),F3),X)),A5)) ) ).

% SUP_apply
tff(fact_3319_UN__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
          & pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B4,X4))) ) ) ).

% UN_iff
tff(fact_3320_UN__I,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),B2: B,B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B4,A3)))
       => pp(aa(set(B),bool,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5)))) ) ) ).

% UN_I
tff(fact_3321_empty__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),bot_bot(set(option(A)))) = none(A) ) ) ).

% empty_Sup
tff(fact_3322_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_3323_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mo(B,A)),A5)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_3324_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B4),A5)) = bot_bot(A) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( aa(B,A,B4,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_3325_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A5: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B4),A5)) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( aa(B,A,B4,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_3326_Sup__multiset__empty,axiom,
    ! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).

% Sup_multiset_empty
tff(fact_3327_Sup__SUP__eq,axiom,
    ! [A: $tType,S: set(fun(A,bool)),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),S),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(A,bool)),set(set(A)),image2(fun(A,bool),set(A),collect(A)),S)))) ) ).

% Sup_SUP_eq
tff(fact_3328_Sup__set__def,axiom,
    ! [A: $tType,A5: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_mp(set(set(A)),fun(A,bool),A5)) ).

% Sup_set_def
tff(fact_3329_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(C,set(product_prod(A,B))),S: set(C),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_mq(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R3)),S)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R3),S)))) ) ).

% SUP_UN_eq2
tff(fact_3330_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool)))),S)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S))) ) ).

% SUP_Sup_eq2
tff(fact_3331_SUP__Sup__eq,axiom,
    ! [A: $tType,S: set(set(A)),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(set(A)),set(fun(A,bool)),image2(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool))),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S))) ) ).

% SUP_Sup_eq
tff(fact_3332_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,bool))),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),S),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S))))) ) ).

% Sup_SUP_eq2
tff(fact_3333_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,set(A)),S: set(B),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),aTP_Lamp_mr(fun(B,set(A)),fun(B,fun(A,bool)),R3)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R3),S)))) ) ).

% SUP_UN_eq
tff(fact_3334_Inf_OINF__identity__eq,axiom,
    ! [A: $tType,Inf: fun(set(A),A),A5: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),image2(A,A,aTP_Lamp_cq(A,A)),A5)) = aa(set(A),A,Inf,A5) ).

% Inf.INF_identity_eq
tff(fact_3335_Sup_OSUP__identity__eq,axiom,
    ! [A: $tType,Sup: fun(set(A),A),A5: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),image2(A,A,aTP_Lamp_cq(A,A)),A5)) = aa(set(A),A,Sup,A5) ).

% Sup.SUP_identity_eq
tff(fact_3336_Sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A5: set(fun(A,B)),X5: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A5),X5) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_ml(A,fun(fun(A,B),B),X5)),A5)) ) ).

% Sup_fun_def
tff(fact_3337_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),X: A] :
          ( ! [Y3: A] :
              ( pp(aa(set(A),bool,member(A,Y3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
         => ( ! [Y3: A] :
                ( ! [Z5: A] :
                    ( pp(aa(set(A),bool,member(A,Z5),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z5),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A5) = X ) ) ) ) ).

% Sup_eqI
tff(fact_3338_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: set(A)] :
          ( ! [A6: A] :
              ( pp(aa(set(A),bool,member(A,A6),A5))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X5)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ).

% Sup_mono
tff(fact_3339_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z2)) ) ) ).

% Sup_least
tff(fact_3340_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,X),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ).

% Sup_upper
tff(fact_3341_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B2))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) ) ) ) ).

% Sup_le_iff
tff(fact_3342_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V2: A] :
          ( pp(aa(set(A),bool,member(A,U),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ).

% Sup_upper2
tff(fact_3343_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,S: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),S))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ).

% less_Sup_iff
tff(fact_3344_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_3345_Union__empty__conv,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) = bot_bot(set(A)) )
    <=> ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),A5))
         => ( X4 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_3346_empty__Union__conv,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) )
    <=> ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),A5))
         => ( X4 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_3347_Union__mono,axiom,
    ! [A: $tType,A5: set(set(A)),B4: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A5),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4))) ) ).

% Union_mono
tff(fact_3348_Union__least,axiom,
    ! [A: $tType,A5: set(set(A)),C4: set(A)] :
      ( ! [X9: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X9),A5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X9),C4)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)),C4)) ) ).

% Union_least
tff(fact_3349_Union__upper,axiom,
    ! [A: $tType,B4: set(A),A5: set(set(A))] :
      ( pp(aa(set(set(A)),bool,member(set(A),B4),A5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5))) ) ).

% Union_upper
tff(fact_3350_Union__subsetI,axiom,
    ! [A: $tType,A5: set(set(A)),B4: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A5))
         => ? [Y4: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Y4),B4))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Y4)) ) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4))) ) ).

% Union_subsetI
tff(fact_3351_SUP__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,fun(C,A)),B4: set(C),A5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aTP_Lamp_ms(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F3),B4)),A5)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(set(B),fun(C,A),aTP_Lamp_mu(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F3),A5)),B4)) ) ).

% SUP_commute
tff(fact_3352_image__Union,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S: set(set(B))] : aa(set(B),set(A),image2(B,A,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),S)) ).

% image_Union
tff(fact_3353_Int__Union2,axiom,
    ! [A: $tType,B4: set(set(A)),A5: 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)),B4)),A5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_mv(set(A),fun(set(A),set(A)),A5)),B4)) ).

% Int_Union2
tff(fact_3354_Int__Union,axiom,
    ! [A: $tType,A5: set(A),B4: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5)),B4)) ).

% Int_Union
tff(fact_3355_UN__UN__flatten,axiom,
    ! [B: $tType,A: $tType,C: $tType,C4: fun(B,set(A)),B4: fun(C,set(B)),A5: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),C4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A5)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_mw(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C4),B4)),A5)) ).

% UN_UN_flatten
tff(fact_3356_UN__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))))
     => ~ ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A5))
           => ~ pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B4,X3))) ) ) ).

% UN_E
tff(fact_3357_UN__extend__simps_I8_J,axiom,
    ! [P7: $tType,O: $tType,B4: fun(O,set(P7)),A5: set(set(O))] : aa(set(set(P7)),set(P7),complete_Sup_Sup(set(P7)),aa(set(set(O)),set(set(P7)),image2(set(O),set(P7),aTP_Lamp_mx(fun(O,set(P7)),fun(set(O),set(P7)),B4)),A5)) = aa(set(set(P7)),set(P7),complete_Sup_Sup(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),B4),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A5))) ).

% UN_extend_simps(8)
tff(fact_3358_UN__extend__simps_I9_J,axiom,
    ! [S7: $tType,R7: $tType,Q8: $tType,C4: fun(R7,set(S7)),B4: fun(Q8,set(R7)),A5: set(Q8)] : aa(set(set(S7)),set(S7),complete_Sup_Sup(set(S7)),aa(set(Q8),set(set(S7)),image2(Q8,set(S7),aa(fun(Q8,set(R7)),fun(Q8,set(S7)),aTP_Lamp_my(fun(R7,set(S7)),fun(fun(Q8,set(R7)),fun(Q8,set(S7))),C4),B4)),A5)) = aa(set(set(S7)),set(S7),complete_Sup_Sup(set(S7)),aa(set(R7),set(set(S7)),image2(R7,set(S7),C4),aa(set(set(R7)),set(R7),complete_Sup_Sup(set(R7)),aa(set(Q8),set(set(R7)),image2(Q8,set(R7),B4),A5)))) ).

% UN_extend_simps(9)
tff(fact_3359_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A5: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A5)))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4)) ) ) ) ) ).

% le_Sup_iff
tff(fact_3360_SUP__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(C,A,G3,X5))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,member(C,J2),B4))
               => ? [X5: B] :
                    ( pp(aa(set(B),bool,member(B,X5),A5))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G3,J2)),aa(B,A,F3,X5))) ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G3),B4)) ) ) ) ) ).

% SUP_eq
tff(fact_3361_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,member(A,V3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V3)) )
         => ( ( A5 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ).

% less_eq_Sup
tff(fact_3362_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ).

% Sup_subset_mono
tff(fact_3363_SUP__eq__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),X: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),I5))
               => ( aa(B,A,F3,I3) = X ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),I5)) = X ) ) ) ) ).

% SUP_eq_const
tff(fact_3364_Sup__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ).

% Sup_union_distrib
tff(fact_3365_Union__disjoint,axiom,
    ! [A: $tType,C4: set(set(A)),A5: 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)),C4)),A5) = bot_bot(set(A)) )
    <=> ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),C4))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),A5) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_3366_Union__Int__subset,axiom,
    ! [A: $tType,A5: set(set(A)),B4: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A5),B4))),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)),A5)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)))) ).

% Union_Int_subset
tff(fact_3367_SUP__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F3: fun(B,A),X: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),X)) )
         => ( ! [Y3: A] :
                ( ! [I: B] :
                    ( pp(aa(set(B),bool,member(B,I),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I)),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_3368_SUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( ! [N5: B] :
              ( pp(aa(set(B),bool,member(B,N5),A5))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N5)),aa(C,A,G3,X5))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G3),B4)))) ) ) ).

% SUP_mono
tff(fact_3369_SUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F3: fun(B,A),U: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),U)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),U)) ) ) ).

% SUP_least
tff(fact_3370_SUP__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A5: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),A5)))) ) ) ).

% SUP_mono'
tff(fact_3371_SUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I2),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)))) ) ) ).

% SUP_upper
tff(fact_3372_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),U))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ).

% SUP_le_iff
tff(fact_3373_SUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A5: set(B),U: A,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I2),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)))) ) ) ) ).

% SUP_upper2
tff(fact_3374_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),Y: A,I2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),Y))
         => ( pp(aa(set(B),bool,member(B,I2),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I2)),Y)) ) ) ) ).

% SUP_lessD
tff(fact_3375_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F3: fun(B,A),A5: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,X4))) ) ) ) ).

% less_SUP_iff
tff(fact_3376_SUP__absorb,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [K: B,I5: set(B),A5: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,K),I5))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,A5,K)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,A5),I5))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,A5),I5)) ) ) ) ).

% SUP_absorb
tff(fact_3377_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),G3: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),A5))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mz(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A5)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_3378_UN__extend__simps_I10_J,axiom,
    ! [V4: $tType,U3: $tType,T: $tType,B4: fun(U3,set(V4)),F3: fun(T,U3),A5: set(T)] : aa(set(set(V4)),set(V4),complete_Sup_Sup(set(V4)),aa(set(T),set(set(V4)),image2(T,set(V4),aa(fun(T,U3),fun(T,set(V4)),aTP_Lamp_na(fun(U3,set(V4)),fun(fun(T,U3),fun(T,set(V4))),B4),F3)),A5)) = aa(set(set(V4)),set(V4),complete_Sup_Sup(set(V4)),aa(set(U3),set(set(V4)),image2(U3,set(V4),B4),aa(set(T),set(U3),image2(T,U3,F3),A5))) ).

% UN_extend_simps(10)
tff(fact_3379_image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),B4: fun(C,set(B)),A5: set(C)] : aa(set(B),set(A),image2(B,A,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_nb(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F3),B4)),A5)) ).

% image_UN
tff(fact_3380_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A5: 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_nc(B,set(A))),A5)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_3381_UN__empty,axiom,
    ! [B: $tType,A: $tType,B4: 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),B4),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_3382_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A5: 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),B4),A5)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
         => ( aa(B,set(A),B4,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_3383_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A5: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5)) = bot_bot(set(A)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
         => ( aa(B,set(A),B4,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_3384_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),I5: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))),B4))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),A5,X4)),B4)) ) ) ).

% UN_subset_iff
tff(fact_3385_UN__upper,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,A3)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5)))) ) ).

% UN_upper
tff(fact_3386_UN__least,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: fun(A,set(B)),C4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,X3)),C4)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5))),C4)) ) ).

% UN_least
tff(fact_3387_UN__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),F3: fun(A,set(B)),G3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,X3)),aa(A,set(B),G3,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),A5))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G3),B4)))) ) ) ).

% UN_mono
tff(fact_3388_UN__extend__simps_I5_J,axiom,
    ! [I6: $tType,J4: $tType,A5: set(I6),B4: fun(J4,set(I6)),C4: set(J4)] : aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),inf_inf(set(I6)),A5),aa(set(set(I6)),set(I6),complete_Sup_Sup(set(I6)),aa(set(J4),set(set(I6)),image2(J4,set(I6),B4),C4))) = aa(set(set(I6)),set(I6),complete_Sup_Sup(set(I6)),aa(set(J4),set(set(I6)),image2(J4,set(I6),aa(fun(J4,set(I6)),fun(J4,set(I6)),aTP_Lamp_nd(set(I6),fun(fun(J4,set(I6)),fun(J4,set(I6))),A5),B4)),C4)) ).

% UN_extend_simps(5)
tff(fact_3389_UN__extend__simps_I4_J,axiom,
    ! [H10: $tType,G: $tType,A5: fun(G,set(H10)),C4: set(G),B4: set(H10)] : aa(set(H10),set(H10),aa(set(H10),fun(set(H10),set(H10)),inf_inf(set(H10)),aa(set(set(H10)),set(H10),complete_Sup_Sup(set(H10)),aa(set(G),set(set(H10)),image2(G,set(H10),A5),C4))),B4) = aa(set(set(H10)),set(H10),complete_Sup_Sup(set(H10)),aa(set(G),set(set(H10)),image2(G,set(H10),aa(set(H10),fun(G,set(H10)),aTP_Lamp_ne(fun(G,set(H10)),fun(set(H10),fun(G,set(H10))),A5),B4)),C4)) ).

% UN_extend_simps(4)
tff(fact_3390_Int__UN__distrib,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A5: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))) = 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_nf(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A5)),I5)) ).

% Int_UN_distrib
tff(fact_3391_Int__UN__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A5: fun(B,set(A)),I5: set(B),B4: fun(C,set(A)),J5: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B4),J5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_nh(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A5),B4),J5)),I5)) ).

% Int_UN_distrib2
tff(fact_3392_UN__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,K),I5))
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A5,K)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5)) ) ) ).

% UN_absorb
tff(fact_3393_UN__Un__distrib,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),B4: fun(B,set(A)),I5: 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_ni(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A5),B4)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),I5))) ).

% UN_Un_distrib
tff(fact_3394_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),B4: fun(B,set(A)),C4: 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_ni(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A5),B4)),C4)) = 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),A5),C4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C4))) ).

% Un_Union_image
tff(fact_3395_UN__extend__simps_I6_J,axiom,
    ! [L5: $tType,K6: $tType,A5: fun(K6,set(L5)),C4: set(K6),B4: set(L5)] : aa(set(L5),set(L5),aa(set(L5),fun(set(L5),set(L5)),minus_minus(set(L5)),aa(set(set(L5)),set(L5),complete_Sup_Sup(set(L5)),aa(set(K6),set(set(L5)),image2(K6,set(L5),A5),C4))),B4) = aa(set(set(L5)),set(L5),complete_Sup_Sup(set(L5)),aa(set(K6),set(set(L5)),image2(K6,set(L5),aa(set(L5),fun(K6,set(L5)),aTP_Lamp_nj(fun(K6,set(L5)),fun(set(L5),fun(K6,set(L5))),A5),B4)),C4)) ).

% UN_extend_simps(6)
tff(fact_3396_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F3: fun(B,A),A5: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),aa(B,A,F3,X4))) ) ) ) ) ).

% le_SUP_iff
tff(fact_3397_SUP__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),C2: A,F3: fun(B,A)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(B,A,F3,I3))) )
           => ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),I5)) = C2 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),I5))
                 => ( aa(B,A,F3,X4) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3398_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B4)))) ) ).

% Sup_inter_less_eq
tff(fact_3399_SUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),B4)))) ) ) ) ).

% SUP_subset_mono
tff(fact_3400_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),C2: A] :
          ( ( ( A5 = bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mj(A,fun(B,A),C2)),A5)) = bot_bot(A) ) )
          & ( ( A5 != bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mj(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% SUP_constant
tff(fact_3401_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_3402_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A5: set(B),B4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M6),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M6),B4))) ) ).

% SUP_union
tff(fact_3403_SUP__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(C,set(B)),A5: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),G3),A5)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_nk(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F3),G3)),A5)) ) ).

% SUP_UNION
tff(fact_3404_UN__extend__simps_I2_J,axiom,
    ! [D: $tType,C: $tType,C4: set(C),A5: fun(C,set(D)),B4: set(D)] :
      ( ( ( C4 = bot_bot(set(C)) )
       => ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A5),C4))),B4) = B4 ) )
      & ( ( C4 != bot_bot(set(C)) )
       => ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A5),C4))),B4) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_mg(fun(C,set(D)),fun(set(D),fun(C,set(D))),A5),B4)),C4)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_3405_UN__extend__simps_I3_J,axiom,
    ! [E: $tType,F: $tType,C4: set(F),A5: set(E),B4: fun(F,set(E))] :
      ( ( ( C4 = bot_bot(set(F)) )
       => ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A5),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),B4),C4))) = A5 ) )
      & ( ( C4 != bot_bot(set(F)) )
       => ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A5),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),B4),C4))) = aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_mh(set(E),fun(fun(F,set(E)),fun(F,set(E))),A5),B4)),C4)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_3406_finite__subset__Union,axiom,
    ! [A: $tType,A5: set(A),B12: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)))
       => ~ ! [F11: set(set(A))] :
              ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F11))
             => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F11),B12))
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F11))) ) ) ) ) ).

% finite_subset_Union
tff(fact_3407_card__UNION,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),A5))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),A5))
           => pp(aa(set(A),bool,finite_finite2(A),X3)) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_nl(set(set(A)),int)),aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_nm(set(set(A)),fun(set(set(A)),bool),A5)))) ) ) ) ).

% card_UNION
tff(fact_3408_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),B4: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),B4))),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_nn(fun(B,A),fun(A,fun(B,A)),F3),A3)),B4)) ) ).

% SUP_inf
tff(fact_3409_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B4)),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_no(A,fun(A,A),A3)),B4)) ) ).

% Sup_inf
tff(fact_3410_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F3: fun(B,A),B4: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),B4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_np(A,fun(fun(B,A),fun(B,A)),A3),F3)),B4)) ) ).

% inf_SUP
tff(fact_3411_INF__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_nq(A,A)),A5)) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ).

% INF_identity_eq
tff(fact_3412_INT__I,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B2: B,B4: fun(A,set(B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B4,X3))) )
     => pp(aa(set(B),bool,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5)))) ) ).

% INT_I
tff(fact_3413_INT__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
         => pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B4,X4))) ) ) ).

% INT_iff
tff(fact_3414_Inf__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A5: set(fun(A,B)),X: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A5),X) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_nr(A,fun(fun(A,B),B),X)),A5)) ) ).

% Inf_apply
tff(fact_3415_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A5) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa: A] :
                  ( pp(aa(set(A),bool,member(A,Xa),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa),X4)) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_3416_SUP2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A3: A,A5: set(A),B4: fun(A,fun(B,fun(C,bool))),B2: B,C2: C] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),B4,A3),B2),C2))
       => pp(aa(C,bool,aa(B,fun(C,bool),aa(set(fun(B,fun(C,bool))),fun(B,fun(C,bool)),complete_Sup_Sup(fun(B,fun(C,bool))),aa(set(A),set(fun(B,fun(C,bool))),image2(A,fun(B,fun(C,bool)),B4),A5)),B2),C2)) ) ) ).

% SUP2_I
tff(fact_3417_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_3418_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,X,Y)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_3419_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,X,Y)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_3420_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_3421_SUP1__I,axiom,
    ! [A: $tType,B: $tType,A3: A,A5: set(A),B4: fun(A,fun(B,bool)),B2: B] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),B4,A3),B2))
       => pp(aa(B,bool,aa(set(fun(B,bool)),fun(B,bool),complete_Sup_Sup(fun(B,bool)),aa(set(A),set(fun(B,bool)),image2(A,fun(B,bool),B4),A5)),B2)) ) ) ).

% SUP1_I
tff(fact_3422_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atMost(A),X)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_3423_cINF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(B),C2: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ly(A,fun(B,A),C2)),A5)) = C2 ) ) ) ).

% cINF_const
tff(fact_3424_INF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F3: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mj(A,fun(B,A),F3)),A5)) = F3 ) ) ) ).

% INF_const
tff(fact_3425_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B))] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),I5))
          & pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A5,X5))) )
     => pp(aa(set(B),bool,finite_finite2(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5)))) ) ).

% finite_INT
tff(fact_3426_INF__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [F3: fun(C,fun(B,A)),A5: set(C),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Inf_Inf(fun(B,A)),aa(set(C),set(fun(B,A)),image2(C,fun(B,A),F3),A5)),X) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_ns(fun(C,fun(B,A)),fun(B,fun(C,A)),F3),X)),A5)) ) ).

% INF_apply
tff(fact_3427_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa: B] :
                  ( pp(aa(set(B),bool,member(B,Xa),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,Xa)),X4)) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_3428_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_nt(fun(B,set(A)),fun(B,set(A)),B4)),A5)) ).

% Compl_INT
tff(fact_3429_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_nt(fun(B,set(A)),fun(B,set(A)),B4)),A5)) ).

% Compl_UN
tff(fact_3430_Inter__subset,axiom,
    ! [A: $tType,A5: set(set(A)),B4: set(A)] :
      ( ! [X9: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X9),A5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X9),B4)) )
     => ( ( A5 != bot_bot(set(set(A))) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),B4)) ) ) ).

% Inter_subset
tff(fact_3431_SUP1__E,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,fun(A,bool)),A5: set(B),B2: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),B4),A5)),B2))
     => ~ ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A5))
           => ~ pp(aa(A,bool,aa(B,fun(A,bool),B4,X3),B2)) ) ) ).

% SUP1_E
tff(fact_3432_SUP2__E,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,fun(A,fun(B,bool))),A5: set(C),B2: A,C2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),B4),A5)),B2),C2))
     => ~ ! [X3: C] :
            ( pp(aa(set(C),bool,member(C,X3),A5))
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B4,X3),B2),C2)) ) ) ).

% SUP2_E
tff(fact_3433_Inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A5: set(fun(A,B)),X5: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A5),X5) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_nr(A,fun(fun(A,B),B),X5)),A5)) ) ).

% Inf_fun_def
tff(fact_3434_Some__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(A,option(A),some(A),aa(set(A),A,complete_Inf_Inf(A),A5)) = aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A5)) ) ).

% Some_Inf
tff(fact_3435_Inter__lower,axiom,
    ! [A: $tType,B4: set(A),A5: set(set(A))] :
      ( pp(aa(set(set(A)),bool,member(set(A),B4),A5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),B4)) ) ).

% Inter_lower
tff(fact_3436_Inter__greatest,axiom,
    ! [A: $tType,A5: set(set(A)),C4: set(A)] :
      ( ! [X9: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X9),A5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),X9)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5))) ) ).

% Inter_greatest
tff(fact_3437_Inter__anti__mono,axiom,
    ! [A: $tType,B4: set(set(A)),A5: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B4),A5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4))) ) ).

% Inter_anti_mono
tff(fact_3438_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),X: A] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,member(A,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),I3)) )
         => ( ! [Y3: A] :
                ( ! [I: A] :
                    ( pp(aa(set(A),bool,member(A,I),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),I)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A5) = X ) ) ) ) ).

% Inf_eqI
tff(fact_3439_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A5: set(A)] :
          ( ! [B5: A] :
              ( pp(aa(set(A),bool,member(A,B5),B4))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B5)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B4))) ) ) ).

% Inf_mono
tff(fact_3440_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,X),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),X)) ) ) ).

% Inf_lower
tff(fact_3441_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V2: A] :
          ( pp(aa(set(A),bool,member(A,U),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),V2)) ) ) ) ).

% Inf_lower2
tff(fact_3442_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A5: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A5)))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X4)) ) ) ) ).

% le_Inf_iff
tff(fact_3443_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),A5))) ) ) ).

% Inf_greatest
tff(fact_3444_Inter__Un__distrib,axiom,
    ! [A: $tType,A5: set(set(A)),B4: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),A5),B4)) = 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)),A5)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4)) ).

% Inter_Un_distrib
tff(fact_3445_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S: set(A),A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S)),A3))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),S))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ).

% Inf_less_iff
tff(fact_3446_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z2: A,X6: set(A)] :
          ( pp(aa(set(A),bool,member(A,Z2),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = Z2 ) ) ) ) ).

% cInf_eq_minimum
tff(fact_3447_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3)) )
         => ( ! [Y3: A] :
                ( ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X5)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),A3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ).

% cInf_eq
tff(fact_3448_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B4: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B4)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),sup_sup(A),A3)),B4)) ) ).

% sup_Inf
tff(fact_3449_INF__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,fun(C,A)),B4: set(C),A5: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aTP_Lamp_nu(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F3),B4)),A5)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(set(B),fun(C,A),aTP_Lamp_nv(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F3),A5)),B4)) ) ).

% INF_commute
tff(fact_3450_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B12: set(set(A)),A5: set(A)] :
      ( ( ( B12 = bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)),A5) = A5 ) )
      & ( ( B12 != bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)),A5) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_mv(set(A),fun(set(A),set(A)),A5)),B12)) ) ) ) ).

% Int_Inter_eq(2)
tff(fact_3451_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,B12: set(set(A)),A5: set(A)] :
      ( ( ( B12 = bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)) = A5 ) )
      & ( ( B12 != bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5)),B12)) ) ) ) ).

% Int_Inter_eq(1)
tff(fact_3452_Un__Inter,axiom,
    ! [A: $tType,A5: set(A),B4: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5)),B4)) ).

% Un_Inter
tff(fact_3453_INT__D,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A5: set(B),A3: B] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))))
     => ( pp(aa(set(B),bool,member(B,A3),A5))
       => pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B4,A3))) ) ) ).

% INT_D
tff(fact_3454_INT__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A5: set(B),A3: B] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))))
     => ( ~ pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B4,A3)))
       => ~ pp(aa(set(B),bool,member(B,A3),A5)) ) ) ).

% INT_E
tff(fact_3455_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),A5: set(B),G3: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G3),B4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_nx(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F3),G3),B4)),A5)) ) ).

% INF_sup_distrib2
tff(fact_3456_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F3: fun(B,A),B4: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),B4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ny(A,fun(fun(B,A),fun(B,A)),A3),F3)),B4)) ) ).

% sup_INF
tff(fact_3457_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B4)),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_nz(A,fun(A,A),A3)),B4)) ) ).

% Inf_sup
tff(fact_3458_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),B4: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),B4))),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_oa(fun(B,A),fun(A,fun(B,A)),F3),A3)),B4)) ) ).

% INF_sup
tff(fact_3459_Some__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,option(A),some(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))) = aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(B),set(option(A)),image2(B,option(A),aTP_Lamp_ob(fun(B,A),fun(B,option(A)),F3)),A5)) ) ).

% Some_INF
tff(fact_3460_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) ) ) ) ) ).

% Inf_le_iff
tff(fact_3461_INF__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: set(C),G3: fun(C,A),F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G3,X5)),aa(B,A,F3,I3))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,member(C,J2),B4))
               => ? [X5: B] :
                    ( pp(aa(set(B),bool,member(B,X5),A5))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X5)),aa(C,A,G3,J2))) ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G3),B4)) ) ) ) ) ).

% INF_eq
tff(fact_3462_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ).

% cInf_greatest
tff(fact_3463_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3)) )
           => ( ! [Y3: A] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X5)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),A3)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_3464_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,member(A,V3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),U)) )
         => ( ( A5 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),U)) ) ) ) ).

% Inf_less_eq
tff(fact_3465_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_le_finite
tff(fact_3466_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z2))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z2)) ) ) ) ) ).

% cInf_lessD
tff(fact_3467_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X3)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_3468_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B4))) ) ) ).

% Inf_superset_mono
tff(fact_3469_INF__eq__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),X: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),I5))
               => ( aa(B,A,F3,I3) = X ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),I5)) = X ) ) ) ) ).

% INF_eq_const
tff(fact_3470_Inf__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B4)) ) ).

% Inf_union_distrib
tff(fact_3471_Inter__Un__subset,axiom,
    ! [A: $tType,A5: set(set(A)),B4: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A5),B4)))) ).

% Inter_Un_subset
tff(fact_3472_INF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),U: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)))) ) ) ).

% INF_greatest
tff(fact_3473_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F3: fun(B,A),A5: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ).

% le_INF_iff
tff(fact_3474_INF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A5: set(B),F3: fun(B,A),U: A] :
          ( pp(aa(set(B),bool,member(B,I2),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),U)) ) ) ) ).

% INF_lower2
tff(fact_3475_INF__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A5: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),A5)))) ) ) ).

% INF_mono'
tff(fact_3476_INF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I2),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(B,A,F3,I2))) ) ) ).

% INF_lower
tff(fact_3477_INF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(B),A5: set(C),F3: fun(C,A),G3: fun(B,A)] :
          ( ! [M5: B] :
              ( pp(aa(set(B),bool,member(B,M5),B4))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X5)),aa(B,A,G3,M5))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,F3),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),B4)))) ) ) ).

% INF_mono
tff(fact_3478_INF__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),X: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F3,I3))) )
         => ( ! [Y3: A] :
                ( ! [I: B] :
                    ( pp(aa(set(B),bool,member(B,I),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),aa(B,A,F3,I))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)) = X ) ) ) ) ).

% INF_eqI
tff(fact_3479_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),A3))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),A3)) ) ) ) ).

% INF_less_iff
tff(fact_3480_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F3: fun(B,A),A5: set(B),I2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))))
         => ( pp(aa(set(B),bool,member(B,I2),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F3,I2))) ) ) ) ).

% less_INF_D
tff(fact_3481_INF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),G3: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),A5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oc(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A5)) ) ).

% INF_inf_distrib
tff(fact_3482_INF__absorb,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [K: B,I5: set(B),A5: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,K),I5))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,A5,K)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,A5),I5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,A5),I5)) ) ) ) ).

% INF_absorb
tff(fact_3483_INT__extend__simps_I10_J,axiom,
    ! [V4: $tType,U3: $tType,T: $tType,B4: fun(U3,set(V4)),F3: fun(T,U3),A5: set(T)] : aa(set(set(V4)),set(V4),complete_Inf_Inf(set(V4)),aa(set(T),set(set(V4)),image2(T,set(V4),aa(fun(T,U3),fun(T,set(V4)),aTP_Lamp_na(fun(U3,set(V4)),fun(fun(T,U3),fun(T,set(V4))),B4),F3)),A5)) = aa(set(set(V4)),set(V4),complete_Inf_Inf(set(V4)),aa(set(U3),set(set(V4)),image2(U3,set(V4),B4),aa(set(T),set(U3),image2(T,U3,F3),A5))) ).

% INT_extend_simps(10)
tff(fact_3484_INT__lower,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5))),aa(A,set(B),B4,A3))) ) ).

% INT_lower
tff(fact_3485_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C4: set(B),B4: fun(A,set(B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),aa(A,set(B),B4,X3))) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5)))) ) ).

% INT_greatest
tff(fact_3486_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),F3: fun(A,set(B)),G3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,X3)),aa(A,set(B),G3,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),B4))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G3),A5)))) ) ) ).

% INT_anti_mono
tff(fact_3487_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A5: fun(B,set(A)),I5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(B,set(A),A5,X4))) ) ) ).

% INT_subset_iff
tff(fact_3488_Int__Inter__image,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),B4: fun(B,set(A)),C4: 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_od(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A5),B4)),C4)) = 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),A5),C4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C4))) ).

% Int_Inter_image
tff(fact_3489_INT__Int__distrib,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),B4: fun(B,set(A)),I5: 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_od(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A5),B4)),I5)) = 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),A5),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),I5))) ).

% INT_Int_distrib
tff(fact_3490_INT__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,K),I5))
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,K)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5)) ) ) ).

% INT_absorb
tff(fact_3491_Un__INT__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A5: fun(B,set(A)),I5: set(B),B4: fun(C,set(A)),J5: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B4),J5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_of(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A5),B4),J5)),I5)) ).

% Un_INT_distrib2
tff(fact_3492_Un__INT__distrib,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A5: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))) = 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_og(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A5)),I5)) ).

% Un_INT_distrib
tff(fact_3493_INT__extend__simps_I6_J,axiom,
    ! [L5: $tType,K6: $tType,A5: fun(K6,set(L5)),C4: set(K6),B4: set(L5)] : aa(set(L5),set(L5),aa(set(L5),fun(set(L5),set(L5)),sup_sup(set(L5)),aa(set(set(L5)),set(L5),complete_Inf_Inf(set(L5)),aa(set(K6),set(set(L5)),image2(K6,set(L5),A5),C4))),B4) = aa(set(set(L5)),set(L5),complete_Inf_Inf(set(L5)),aa(set(K6),set(set(L5)),image2(K6,set(L5),aa(set(L5),fun(K6,set(L5)),aTP_Lamp_oh(fun(K6,set(L5)),fun(set(L5),fun(K6,set(L5))),A5),B4)),C4)) ).

% INT_extend_simps(6)
tff(fact_3494_INT__extend__simps_I7_J,axiom,
    ! [M10: $tType,N9: $tType,A5: set(M10),B4: fun(N9,set(M10)),C4: set(N9)] : aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),sup_sup(set(M10)),A5),aa(set(set(M10)),set(M10),complete_Inf_Inf(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),B4),C4))) = aa(set(set(M10)),set(M10),complete_Inf_Inf(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_oi(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),A5),B4)),C4)) ).

% INT_extend_simps(7)
tff(fact_3495_INT__extend__simps_I9_J,axiom,
    ! [S7: $tType,R7: $tType,Q8: $tType,C4: fun(R7,set(S7)),B4: fun(Q8,set(R7)),A5: set(Q8)] : aa(set(set(S7)),set(S7),complete_Inf_Inf(set(S7)),aa(set(Q8),set(set(S7)),image2(Q8,set(S7),aa(fun(Q8,set(R7)),fun(Q8,set(S7)),aTP_Lamp_oj(fun(R7,set(S7)),fun(fun(Q8,set(R7)),fun(Q8,set(S7))),C4),B4)),A5)) = aa(set(set(S7)),set(S7),complete_Inf_Inf(set(S7)),aa(set(R7),set(set(S7)),image2(R7,set(S7),C4),aa(set(set(R7)),set(R7),complete_Sup_Sup(set(R7)),aa(set(Q8),set(set(R7)),image2(Q8,set(R7),B4),A5)))) ).

% INT_extend_simps(9)
tff(fact_3496_INT__extend__simps_I8_J,axiom,
    ! [P7: $tType,O: $tType,B4: fun(O,set(P7)),A5: set(set(O))] : aa(set(set(P7)),set(P7),complete_Inf_Inf(set(P7)),aa(set(set(O)),set(set(P7)),image2(set(O),set(P7),aTP_Lamp_ok(fun(O,set(P7)),fun(set(O),set(P7)),B4)),A5)) = aa(set(set(P7)),set(P7),complete_Inf_Inf(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),B4),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A5))) ).

% INT_extend_simps(8)
tff(fact_3497_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),Y5)) ) ) ) ) ).

% INF_le_iff
tff(fact_3498_cINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(B),M: A,F3: fun(B,A)] :
          ( ( A5 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)))) ) ) ) ).

% cINF_greatest
tff(fact_3499_INF__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),C2: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),C2)) )
           => ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),I5)) = C2 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),I5))
                 => ( aa(B,A,F3,X4) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_3500_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_3501_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ).

% Inf_le_Sup
tff(fact_3502_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A5))
                 => ( pp(aa(set(A),bool,member(A,Y3),A5))
                   => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)),A5)) ) )
             => pp(aa(set(A),bool,member(A,aa(set(A),A,complete_Inf_Inf(A),A5)),A5)) ) ) ) ) ).

% finite_Inf_in
tff(fact_3503_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S))),A3)) ) ) ) ).

% cInf_abs_ge
tff(fact_3504_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B4))),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)),A5),B4)))) ) ).

% less_eq_Inf_inter
tff(fact_3505_INF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(B),A5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),B4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),B4)))) ) ) ) ).

% INF_superset_mono
tff(fact_3506_INF__inf__const1,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),X: A,F3: fun(B,A)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ol(A,fun(fun(B,A),fun(B,A)),X),F3)),I5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_3507_INF__inf__const2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),X: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_om(fun(B,A),fun(A,fun(B,A)),F3),X)),I5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),I5))),X) ) ) ) ).

% INF_inf_const2
tff(fact_3508_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B4: fun(B,A),A5: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B4),A5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_on(fun(B,A),fun(B,A),B4)),A5)) ) ).

% uminus_SUP
tff(fact_3509_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B4: fun(B,A),A5: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B4),A5))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_on(fun(B,A),fun(B,A),B4)),A5)) ) ).

% uminus_INF
tff(fact_3510_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A5: set(B),B4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M6),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M6),B4))) ) ).

% INF_union
tff(fact_3511_INT__extend__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C4: set(D),A5: set(C),B4: fun(D,set(C))] :
      ( ( ( C4 = bot_bot(set(D)) )
       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A5),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),C4))) = A5 ) )
      & ( ( C4 != bot_bot(set(D)) )
       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A5),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),C4))) = aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_oo(set(C),fun(fun(D,set(C)),fun(D,set(C))),A5),B4)),C4)) ) ) ) ).

% INT_extend_simps(2)
tff(fact_3512_INT__extend__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,C4: set(A),A5: fun(A,set(B)),B4: set(B)] :
      ( ( ( C4 = bot_bot(set(A)) )
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),C4))),B4) = B4 ) )
      & ( ( C4 != bot_bot(set(A)) )
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),C4))),B4) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_op(fun(A,set(B)),fun(set(B),fun(A,set(B))),A5),B4)),C4)) ) ) ) ).

% INT_extend_simps(1)
tff(fact_3513_INT__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A5: set(B),B4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))) = 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),M6),A5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),B4))) ).

% INT_Un
tff(fact_3514_UN__extend__simps_I7_J,axiom,
    ! [M10: $tType,N9: $tType,A5: set(M10),B4: fun(N9,set(M10)),C4: set(N9)] : aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),minus_minus(set(M10)),A5),aa(set(set(M10)),set(M10),complete_Inf_Inf(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),B4),C4))) = aa(set(set(M10)),set(M10),complete_Sup_Sup(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_oq(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),A5),B4)),C4)) ).

% UN_extend_simps(7)
tff(fact_3515_INF__le__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( ( A5 != bot_bot(set(B)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)))) ) ) ).

% INF_le_SUP
tff(fact_3516_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S)),L))),E3)) ) ) ) ).

% cInf_asclose
tff(fact_3517_INT__extend__simps_I4_J,axiom,
    ! [G: $tType,H10: $tType,C4: set(H10),A5: set(G),B4: fun(H10,set(G))] :
      ( ( ( C4 = bot_bot(set(H10)) )
       => ( aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),A5),aa(set(set(G)),set(G),complete_Sup_Sup(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),B4),C4))) = A5 ) )
      & ( ( C4 != bot_bot(set(H10)) )
       => ( aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),A5),aa(set(set(G)),set(G),complete_Sup_Sup(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),B4),C4))) = aa(set(set(G)),set(G),complete_Inf_Inf(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_or(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),A5),B4)),C4)) ) ) ) ).

% INT_extend_simps(4)
tff(fact_3518_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B4)),A3) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),B4))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A3) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3519_inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B4: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),B4)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),inf_inf(A),A3)),B4)) ) ).

% inf_Sup
tff(fact_3520_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),A5: set(B),G3: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G3),B4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ot(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F3),G3),B4)),A5)) ) ).

% SUP_inf_distrib2
tff(fact_3521_UN__UN__split__split__eq,axiom,
    ! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,A5: fun(B,fun(C,fun(D,fun(E,set(A))))),Y6: set(product_prod(D,E)),X6: set(product_prod(B,C))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image2(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(set(product_prod(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_ou(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),A5),Y6))),X6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image2(product_prod(B,C),set(A),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_ox(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),A5),Y6)),X6)) ).

% UN_UN_split_split_eq
tff(fact_3522_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F3: fun(nat,set(A)),S: set(A)] :
      ( ! [I3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F3,I3)),S))
     => ( pp(aa(set(A),bool,finite_finite2(A),S))
       => ( ? [N10: nat] :
              ( ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),N10))
                 => ! [M5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N10))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N5))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F3,M5)),aa(nat,set(A),F3,N5))) ) ) )
              & ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N10),N5))
                 => ( aa(nat,set(A),F3,N10) = aa(nat,set(A),F3,N5) ) ) )
         => ( aa(nat,set(A),F3,aa(set(A),nat,finite_card(A),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F3),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_3523_UN__constant__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,A5: set(A),F3: fun(A,set(B)),C2: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => ( aa(A,set(B),F3,X3) = C2 ) )
       => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),A5)) = C2 ) ) ) ).

% UN_constant_eq
tff(fact_3524_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A5)),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),finite_Fpow(B,A5))),finite_Fpow(A,B4))) ) ).

% image_Fpow_mono
tff(fact_3525_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( ( L = zero_zero(code_integer) )
           => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
          & ( ( L != zero_zero(code_integer) )
           => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_oy(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_3526_top__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( top(C)
     => ! [X: D] : aa(D,C,top_top(fun(D,C)),X) = top_top(C) ) ).

% top_apply
tff(fact_3527_UNIV__I,axiom,
    ! [A: $tType,X: A] : pp(aa(set(A),bool,member(A,X),top_top(set(A)))) ).

% UNIV_I
tff(fact_3528_INF2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: fun(A,fun(B,fun(C,bool))),B2: B,C2: C] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),B4,X3),B2),C2)) )
     => pp(aa(C,bool,aa(B,fun(C,bool),aa(set(fun(B,fun(C,bool))),fun(B,fun(C,bool)),complete_Inf_Inf(fun(B,fun(C,bool))),aa(set(A),set(fun(B,fun(C,bool))),image2(A,fun(B,fun(C,bool)),B4),A5)),B2),C2)) ) ).

% INF2_I
tff(fact_3529_INF1__I,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: fun(A,fun(B,bool)),B2: B] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => pp(aa(B,bool,aa(A,fun(B,bool),B4,X3),B2)) )
     => pp(aa(B,bool,aa(set(fun(B,bool)),fun(B,bool),complete_Inf_Inf(fun(B,bool)),aa(set(A),set(fun(B,bool)),image2(A,fun(B,bool),B4),A5)),B2)) ) ).

% INF1_I
tff(fact_3530_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),top_top(set(option(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_option_UNIV
tff(fact_3531_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_3532_inf__top_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B2: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
        <=> ( ( A3 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_3533_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_3534_inf__top_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = top_top(A) )
        <=> ( ( A3 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_3535_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A,Y: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) )
        <=> ( ( X = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_3536_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = top_top(A) )
        <=> ( ( X = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_3537_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).

% inf_top_right
tff(fact_3538_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),X) = X ) ).

% inf_top_left
tff(fact_3539_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_3540_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% sup_top_left
tff(fact_3541_boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% boolean_algebra.disj_one_right
tff(fact_3542_boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% boolean_algebra.disj_one_left
tff(fact_3543_Int__UNIV,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = top_top(set(A)) )
    <=> ( ( A5 = top_top(set(A)) )
        & ( B4 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_3544_card__eq__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),S) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV
tff(fact_3545_card__eq__UNIV2,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) = aa(set(A),nat,finite_card(A),S) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV2
tff(fact_3546_Collect__const,axiom,
    ! [A: $tType,P: bool] :
      ( ( pp(P)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_oz(bool,fun(A,bool),P)) = top_top(set(A)) ) )
      & ( ~ pp(P)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_oz(bool,fun(A,bool),P)) = bot_bot(set(A)) ) ) ) ).

% Collect_const
tff(fact_3547_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ci(fun(A,bool),fun(A,bool),P))))
      <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_Collect_not
tff(fact_3548_subset__mset_OcINF__const,axiom,
    ! [B: $tType,A: $tType,A5: set(B),C2: multiset(A)] :
      ( ( A5 != bot_bot(set(B)) )
     => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aTP_Lamp_mk(multiset(A),fun(B,multiset(A)),C2)),A5)) = C2 ) ) ).

% subset_mset.cINF_const
tff(fact_3549_None__in__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(option(A))] :
          ( pp(aa(set(option(A)),bool,member(option(A),none(A)),A5))
         => ( aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),A5) = none(A) ) ) ) ).

% None_in_Inf
tff(fact_3550_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A5) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa: A] :
                  ( pp(aa(set(A),bool,member(A,Xa),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Xa)) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_3551_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_3552_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_3553_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = top_top(A) ) ).

% sup_compl_top_left1
tff(fact_3554_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y)) = top_top(A) ) ).

% sup_compl_top_left2
tff(fact_3555_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_3556_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_3557_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_3558_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_3559_Diff__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_3560_card__ge__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(A),nat,finite_card(A),S)))
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_ge_UNIV
tff(fact_3561_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cz(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_3562_INF__top__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A5: set(B)] :
          ( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B4),A5)) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( aa(B,A,B4,X4) = top_top(A) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_3563_INF__top__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B4),A5)) = top_top(A) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( aa(B,A,B4,X4) = top_top(A) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_3564_INF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_pa(B,A)),A5)) = top_top(A) ) ).

% INF_top
tff(fact_3565_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa: B] :
                  ( pp(aa(set(B),bool,member(B,Xa),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),aa(B,A,F3,Xa))) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_3566_INT__constant,axiom,
    ! [B: $tType,A: $tType,A5: set(B),C2: set(A)] :
      ( ( ( A5 = bot_bot(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_mi(set(A),fun(B,set(A)),C2)),A5)) = top_top(set(A)) ) )
      & ( ( A5 != bot_bot(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_mi(set(A),fun(B,set(A)),C2)),A5)) = C2 ) ) ) ).

% INT_constant
tff(fact_3567_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),X)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_3568_INT__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C4: set(D),A5: set(C),B4: fun(D,set(C))] :
      ( ( ( C4 = bot_bot(set(D)) )
       => ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_oo(set(C),fun(fun(D,set(C)),fun(D,set(C))),A5),B4)),C4)) = top_top(set(C)) ) )
      & ( ( C4 != bot_bot(set(D)) )
       => ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_oo(set(C),fun(fun(D,set(C)),fun(D,set(C))),A5),B4)),C4)) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A5),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),C4))) ) ) ) ).

% INT_simps(2)
tff(fact_3569_INT__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C4: set(A),A5: fun(A,set(B)),B4: set(B)] :
      ( ( ( C4 = bot_bot(set(A)) )
       => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_op(fun(A,set(B)),fun(set(B),fun(A,set(B))),A5),B4)),C4)) = top_top(set(B)) ) )
      & ( ( C4 != bot_bot(set(A)) )
       => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_op(fun(A,set(B)),fun(set(B),fun(A,set(B))),A5),B4)),C4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),C4))),B4) ) ) ) ).

% INT_simps(1)
tff(fact_3570_INT__simps_I3_J,axiom,
    ! [E: $tType,F: $tType,C4: set(E),A5: fun(E,set(F)),B4: set(F)] :
      ( ( ( C4 = bot_bot(set(E)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_pb(fun(E,set(F)),fun(set(F),fun(E,set(F))),A5),B4)),C4)) = top_top(set(F)) ) )
      & ( ( C4 != bot_bot(set(E)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_pb(fun(E,set(F)),fun(set(F),fun(E,set(F))),A5),B4)),C4)) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),A5),C4))),B4) ) ) ) ).

% INT_simps(3)
tff(fact_3571_INT__simps_I4_J,axiom,
    ! [G: $tType,H10: $tType,C4: set(H10),A5: set(G),B4: fun(H10,set(G))] :
      ( ( ( C4 = bot_bot(set(H10)) )
       => ( aa(set(set(G)),set(G),complete_Inf_Inf(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_or(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),A5),B4)),C4)) = top_top(set(G)) ) )
      & ( ( C4 != bot_bot(set(H10)) )
       => ( aa(set(set(G)),set(G),complete_Inf_Inf(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_or(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),A5),B4)),C4)) = aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),A5),aa(set(set(G)),set(G),complete_Sup_Sup(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),B4),C4))) ) ) ) ).

% INT_simps(4)
tff(fact_3572_surj__fun__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(B,A),X6: set(B),G1: fun(A,C),G22: fun(A,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),X6) = top_top(set(A)) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),X6))
           => ( aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G1),F3),X3) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G22),F3),X3) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_3573_Inf__nat__def1,axiom,
    ! [K5: set(nat)] :
      ( ( K5 != bot_bot(set(nat)) )
     => pp(aa(set(nat),bool,member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K5)),K5)) ) ).

% Inf_nat_def1
tff(fact_3574_rangeI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: B] : pp(aa(set(A),bool,member(A,aa(B,A,F3,X)),aa(set(B),set(A),image2(B,A,F3),top_top(set(B))))) ).

% rangeI
tff(fact_3575_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),X: B] :
      ( ( B2 = aa(B,A,F3,X) )
     => pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),top_top(set(B))))) ) ).

% range_eqI
tff(fact_3576_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_3577_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bm(A,bool)) ).

% UNIV_def
tff(fact_3578_boolean__algebra_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).

% boolean_algebra.conj_one_right
tff(fact_3579_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),top_top(A))) ) ).

% top_greatest
tff(fact_3580_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_3581_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_3582_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [X: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,X,Y) = top_top(set(A)) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_3583_subset__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),top_top(set(A)))) ).

% subset_UNIV
tff(fact_3584_Int__UNIV__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B4) = B4 ).

% Int_UNIV_left
tff(fact_3585_Int__UNIV__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),top_top(set(A))) = A5 ).

% Int_UNIV_right
tff(fact_3586_top__option__def,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ( top_top(option(A)) = aa(A,option(A),some(A),top_top(A)) ) ) ).

% top_option_def
tff(fact_3587_comp__cong,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F3: fun(B,A),G3: fun(C,B),X: C,F12: fun(D,A),G6: fun(E,D),X8: E] :
      ( ( aa(B,A,F3,aa(C,B,G3,X)) = aa(D,A,F12,aa(E,D,G6,X8)) )
     => ( aa(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3),X) = aa(E,A,aa(fun(E,D),fun(E,A),comp(D,A,E,F12),G6),X8) ) ) ).

% comp_cong
tff(fact_3588_UNIV__eq__I,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X3: A] : pp(aa(set(A),bool,member(A,X3),A5))
     => ( top_top(set(A)) = A5 ) ) ).

% UNIV_eq_I
tff(fact_3589_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X3: A] : pp(aa(set(A),bool,member(A,X3),top_top(set(A)))) ).

% UNIV_witness
tff(fact_3590_comp__cong__left,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: fun(A,B),Y: fun(A,B),F3: fun(C,A)] :
      ( ( X = Y )
     => ( aa(fun(C,A),fun(C,B),comp(A,B,C,X),F3) = aa(fun(C,A),fun(C,B),comp(A,B,C,Y),F3) ) ) ).

% comp_cong_left
tff(fact_3591_comp__cong__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,B),Y: fun(A,B),F3: fun(B,C)] :
      ( ( X = Y )
     => ( aa(fun(A,B),fun(A,C),comp(B,C,A,F3),X) = aa(fun(A,B),fun(A,C),comp(B,C,A,F3),Y) ) ) ).

% comp_cong_right
tff(fact_3592_fun__comp__eq__conv,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(C,B),G3: fun(A,C),Fg: fun(A,B)] :
      ( ( aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3) = Fg )
    <=> ! [X4: A] : aa(C,B,F3,aa(A,C,G3,X4)) = aa(A,B,Fg,X4) ) ).

% fun_comp_eq_conv
tff(fact_3593_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A3)) ) ).

% top.extremum_strict
tff(fact_3594_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),top_top(A))) ) ) ).

% top.not_eq_extremum
tff(fact_3595_Un__UNIV__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_3596_Un__UNIV__left,axiom,
    ! [A: $tType,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B4) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_3597_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(C,B),G3: fun(D,C),X: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G3),X)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),comp(C,B,D,F3),G3)),X) ).

% apsnd_compose
tff(fact_3598_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),G3: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pc(fun(C,A),fun(fun(B,C),fun(B,A)),F3),G3)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F3),aa(set(B),set(C),image2(B,C,G3),top_top(set(B)))) ).

% range_composition
tff(fact_3599_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))))
     => ~ ! [X3: B] : B2 != aa(B,A,F3,X3) ) ).

% rangeE
tff(fact_3600_INF2__D,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,fun(A,fun(B,bool))),A5: set(C),B2: A,C2: B,A3: C] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),B4),A5)),B2),C2))
     => ( pp(aa(set(C),bool,member(C,A3),A5))
       => pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B4,A3),B2),C2)) ) ) ).

% INF2_D
tff(fact_3601_INF2__E,axiom,
    ! [B: $tType,A: $tType,C: $tType,B4: fun(C,fun(A,fun(B,bool))),A5: set(C),B2: A,C2: B,A3: C] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),B4),A5)),B2),C2))
     => ( ~ pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B4,A3),B2),C2))
       => ~ pp(aa(set(C),bool,member(C,A3),A5)) ) ) ).

% INF2_E
tff(fact_3602_INF1__D,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,fun(A,bool)),A5: set(B),B2: A,A3: B] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),B4),A5)),B2))
     => ( pp(aa(set(B),bool,member(B,A3),A5))
       => pp(aa(A,bool,aa(B,fun(A,bool),B4,A3),B2)) ) ) ).

% INF1_D
tff(fact_3603_INF1__E,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,fun(A,bool)),A5: set(B),B2: A,A3: B] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),B4),A5)),B2))
     => ( ~ pp(aa(A,bool,aa(B,fun(A,bool),B4,A3),B2))
       => ~ pp(aa(set(B),bool,member(B,A3),A5)) ) ) ).

% INF1_E
tff(fact_3604_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_3605_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_3606_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B2)) = top_top(A) ) ).

% sup_cancel_left1
tff(fact_3607_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B2)) = top_top(A) ) ).

% sup_cancel_left2
tff(fact_3608_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B4)),A3) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),B4))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A3) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_3609_empty__in__Fpow,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(set(A)),bool,member(set(A),bot_bot(set(A))),finite_Fpow(A,A5))) ).

% empty_in_Fpow
tff(fact_3610_Inf__multiset__empty,axiom,
    ! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).

% Inf_multiset_empty
tff(fact_3611_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),B4: set(A),I2: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))),B4))
     => pp(aa(set(A),bool,member(A,aa(B,A,F3,I2)),B4)) ) ).

% range_subsetD
tff(fact_3612_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_3613_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).

% not_UNIV_le_Icc
tff(fact_3614_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H))) ) ).

% not_UNIV_le_Iic
tff(fact_3615_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_3616_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_3617_INF__filter__not__bot,axiom,
    ! [I6: $tType,A: $tType,B4: set(I6),F4: fun(I6,filter(A))] :
      ( ! [X9: set(I6)] :
          ( pp(aa(set(I6),bool,aa(set(I6),fun(set(I6),bool),ord_less_eq(set(I6)),X9),B4))
         => ( pp(aa(set(I6),bool,finite_finite2(I6),X9))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image2(I6,filter(A),F4),X9)) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image2(I6,filter(A),F4),B4)) != bot_bot(filter(A)) ) ) ).

% INF_filter_not_bot
tff(fact_3618_Compl__partition2,axiom,
    ! [A: $tType,A5: 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)),A5)),A5) = top_top(set(A)) ).

% Compl_partition2
tff(fact_3619_Compl__partition,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = top_top(set(A)) ).

% Compl_partition
tff(fact_3620_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A5) ).

% Compl_eq_Diff_UNIV
tff(fact_3621_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_pe(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(fun(B,C)),set(A),image2(fun(B,C),A,aTP_Lamp_pg(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_3622_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ph(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,C)),set(A),image2(fun(B,C),A,aTP_Lamp_pi(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_3623_Fpow__not__empty,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_3624_Inter__empty,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),bot_bot(set(set(A)))) = top_top(set(A)) ).

% Inter_empty
tff(fact_3625_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G3: fun(B,A),F3: fun(A,C)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,G3),top_top(set(B)))))
     => pp(aa(set(C),bool,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_pj(fun(B,A),fun(fun(A,C),fun(B,C)),G3),F3)),top_top(set(B))))) ) ).

% finite_range_imageI
tff(fact_3626_Inf__INT__eq,axiom,
    ! [A: $tType,S: set(fun(A,bool)),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),S),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(fun(A,bool)),set(set(A)),image2(fun(A,bool),set(A),collect(A)),S)))) ) ).

% Inf_INT_eq
tff(fact_3627_INTER__UNIV__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A5: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5)) = top_top(set(A)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
         => ( aa(B,set(A),B4,X4) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_3628_INTER__UNIV__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A5: set(B)] :
      ( ( top_top(set(A)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
         => ( aa(B,set(A),B4,X4) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_3629_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% sup_shunt
tff(fact_3630_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A3: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),X) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),X) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Y) = top_top(A) )
               => ( X = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_3631_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_3632_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_3633_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_3634_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool)))),S)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S))) ) ).

% INF_Int_eq2
tff(fact_3635_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_3636_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),C2: A] :
          ( ( ( A5 = bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mj(A,fun(B,A),C2)),A5)) = top_top(A) ) )
          & ( ( A5 != bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mj(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% INF_constant
tff(fact_3637_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),uminus_uminus(set(B)),A5)))) ) ).

% surj_Compl_image_subset
tff(fact_3638_INF__Int__eq,axiom,
    ! [A: $tType,S: set(set(A)),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(set(A)),set(fun(A,bool)),image2(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool))),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S))) ) ).

% INF_Int_eq
tff(fact_3639_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,set(A)),S: set(B),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),aTP_Lamp_mr(fun(B,set(A)),fun(B,fun(A,bool)),R3)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R3),S)))) ) ).

% INF_INT_eq
tff(fact_3640_finite__range__Some,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_range_Some
tff(fact_3641_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(C,set(product_prod(A,B))),S: set(C),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_mq(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R3)),S)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R3),S)))) ) ).

% INF_INT_eq2
tff(fact_3642_Inf__set__def,axiom,
    ! [A: $tType,A5: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_pk(set(set(A)),fun(A,bool),A5)) ).

% Inf_set_def
tff(fact_3643_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ pp(aa(set(option(A)),bool,member(option(A),X),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_3644_INT__empty,axiom,
    ! [B: $tType,A: $tType,B4: 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),B4),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_3645_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),X) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_3646_Fpow__mono,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A5)),finite_Fpow(A,B4))) ) ).

% Fpow_mono
tff(fact_3647_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_3648_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).

% finite_UNIV_card_ge_0
tff(fact_3649_Fpow__def,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_pl(set(A),fun(set(A),bool),A5)) ).

% Fpow_def
tff(fact_3650_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(C),G3: fun(A,B),F3: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite2(C),I5))
         => ( ! [I3: C] :
                ( pp(aa(set(C),bool,member(C,I3),I5))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G3,aa(C,A,F3,I3)))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(C),set(A),image2(C,A,F3),I5))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),comp(A,B,C,G3),F3)),I5))) ) ) ) ).

% sum_image_le
tff(fact_3651_INT__extend__simps_I3_J,axiom,
    ! [F: $tType,E: $tType,C4: set(E),A5: fun(E,set(F)),B4: set(F)] :
      ( ( ( C4 = bot_bot(set(E)) )
       => ( aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),A5),C4))),B4) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),top_top(set(F))),B4) ) )
      & ( ( C4 != bot_bot(set(E)) )
       => ( aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),A5),C4))),B4) = aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_pb(fun(E,set(F)),fun(set(F),fun(E,set(F))),A5),B4)),C4)) ) ) ) ).

% INT_extend_simps(3)
tff(fact_3652_range__mod,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_pm(nat,fun(nat,nat),N2)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N2) ) ) ).

% range_mod
tff(fact_3653_UN__UN__finite__eq,axiom,
    ! [A: $tType,A5: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aTP_Lamp_pn(fun(nat,set(A)),fun(nat,set(A)),A5)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_3654_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))))) ) ).

% card_range_greater_zero
tff(fact_3655_UN__finite__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),C4: set(A)] :
      ( ! [N5: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N5)))),C4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),top_top(set(nat))))),C4)) ) ).

% UN_finite_subset
tff(fact_3656_UN__finite2__eq,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N5: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B4),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_3657_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,bool))),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),S),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S))))) ) ).

% Inf_INT_eq2
tff(fact_3658_UN__finite2__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N5: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N5)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),K))))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B4),top_top(set(nat)))))) ) ).

% UN_finite2_subset
tff(fact_3659_Inf__option__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(option(A))] :
          ( ( pp(aa(set(option(A)),bool,member(option(A),none(A)),A5))
           => ( aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),A5) = none(A) ) )
          & ( ~ pp(aa(set(option(A)),bool,member(option(A),none(A)),A5))
           => ( aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),A5) = aa(A,option(A),some(A),aa(set(A),A,complete_Inf_Inf(A),these(A,A5))) ) ) ) ) ).

% Inf_option_def
tff(fact_3660_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_3661_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N2: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N2),A3) = bit_se1065995026697491101ons_or(A,aa(A,A,bit_se2584673776208193580ke_bit(A,N2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N2))),bit_ri4277139882892585799ns_not(A,bit_se2239418461657761734s_mask(A,N2)))) ) ).

% signed_take_bit_def
tff(fact_3662_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_po(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_3663_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit0,N2)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_pp(num,option(num)),bit_and_not_num(M,N2)) ).

% and_not_num.simps(8)
tff(fact_3664_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_3665_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_3666_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_3667_Suc__funpow,axiom,
    ! [N2: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N2),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N2) ).

% Suc_funpow
tff(fact_3668_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344872417868541ns_and(int,K,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_3669_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344872417868541ns_and(int,K,L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% and_negative_int_iff
tff(fact_3670_funpow__0,axiom,
    ! [A: $tType,F3: fun(A,A),X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3),X) = X ).

% funpow_0
tff(fact_3671_mod__true,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(top_top(assn)),H))
    <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H)) ) ).

% mod_true
tff(fact_3672_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: bool] :
      ( ( pp(P)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_pq(bool,fun(A,fun(B,bool)),P))) = top_top(set(product_prod(A,B))) ) )
      & ( ~ pp(P)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_pq(bool,fun(A,fun(B,bool)),P))) = bot_bot(set(product_prod(A,B))) ) ) ) ).

% Collect_const_case_prod
tff(fact_3673_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_ri4277139882892585799ns_not(int,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% not_negative_int_iff
tff(fact_3674_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_ri4277139882892585799ns_not(int,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% not_nonnegative_int_iff
tff(fact_3675_surj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N2: nat] :
      ( ( aa(set(A),set(A),image2(A,A,F3),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image2(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_3676_these__image__Some__eq,axiom,
    ! [A: $tType,A5: set(A)] : these(A,aa(set(A),set(option(A)),image2(A,option(A),some(A)),A5)) = A5 ).

% these_image_Some_eq
tff(fact_3677_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_3678_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_3679_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_3680_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_3681_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] : bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N2)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N2))) ).

% and_minus_numerals(4)
tff(fact_3682_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] : bit_se5824344872417868541ns_and(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N2))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N2))) ).

% and_minus_numerals(8)
tff(fact_3683_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] : bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N2)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N2))) ).

% and_minus_numerals(3)
tff(fact_3684_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] : bit_se5824344872417868541ns_and(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N2))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N2))) ).

% and_minus_numerals(7)
tff(fact_3685_in__Union__o__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: A,Gset: fun(B,set(set(A))),Gmap: fun(C,B),A5: C] :
      ( pp(aa(set(A),bool,member(A,X),aa(C,set(A),aa(fun(C,B),fun(C,set(A)),comp(B,set(A),C,aa(fun(B,set(set(A))),fun(B,set(A)),comp(set(set(A)),set(A),B,complete_Sup_Sup(set(A))),Gset)),Gmap),A5)))
     => pp(aa(set(A),bool,member(A,X),aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A))),aa(fun(C,B),fun(C,set(set(A))),comp(B,set(set(A)),C,Gset),Gmap)),A5))) ) ).

% in_Union_o_assoc
tff(fact_3686_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F4: fun(A,filter(B))] :
      ( ! [I3: A] :
          ( pp(aa(set(A),bool,member(A,I3),I5))
         => ! [J2: A] :
              ( pp(aa(set(A),bool,member(A,J2),I5))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),I5))
                  & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F4,X5)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I3)),aa(A,filter(B),F4,J2)))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I5)) = bot_bot(filter(B)) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),I5))
            & ( aa(A,filter(B),F4,X4) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_3687_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(A,C),G3: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_pr(C,set(B))),F3) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image2(D,B,G3)),aTP_Lamp_ps(A,set(D))) ).

% empty_natural
tff(fact_3688_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X5),Xa3))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),top_top(set(product_prod(A,B))))) ) ).

% top_empty_eq2
tff(fact_3689_Inf__filter__not__bot,axiom,
    ! [A: $tType,B4: set(filter(A))] :
      ( ! [X9: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X9),B4))
         => ( pp(aa(set(filter(A)),bool,finite_finite2(filter(A)),X9))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X9) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B4) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_3690_comp__funpow,axiom,
    ! [B: $tType,A: $tType,N2: nat,F3: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N2),comp(A,A,B,F3)) = comp(A,A,B,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3)) ).

% comp_funpow
tff(fact_3691_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,bool),set(A),collect(A),top_top(fun(A,bool))) ).

% top_set_def
tff(fact_3692_top__empty__eq,axiom,
    ! [A: $tType,X5: A] :
      ( pp(aa(A,bool,top_top(fun(A,bool)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),top_top(set(A)))) ) ).

% top_empty_eq
tff(fact_3693_ent__true,axiom,
    ! [P: assn] : entails(P,top_top(assn)) ).

% ent_true
tff(fact_3694_funpow__swap1,axiom,
    ! [A: $tType,F3: fun(A,A),N2: nat,X: A] : aa(A,A,F3,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),aa(A,A,F3,X)) ).

% funpow_swap1
tff(fact_3695_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q3: num] :
      ( ( bit_and_not_num(M,N2) = aa(num,option(num),some(num),Q3) )
    <=> ( bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),N2))) = aa(num,int,numeral_numeral(int),Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_3696_funpow__mult,axiom,
    ! [A: $tType,N2: nat,M: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),F3) ).

% funpow_mult
tff(fact_3697_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] : bit_se5824344872417868541ns_and(int,bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N2)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N2,M)) ).

% int_numeral_not_and_num
tff(fact_3698_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] : bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),N2))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N2)) ).

% int_numeral_and_not_num
tff(fact_3699_Union__natural,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : aa(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B)),comp(set(set(B)),set(B),set(set(A)),complete_Sup_Sup(set(B))),image2(set(A),set(B),image2(A,B,F3))) = aa(fun(set(set(A)),set(A)),fun(set(set(A)),set(B)),comp(set(A),set(B),set(set(A)),image2(A,B,F3)),complete_Sup_Sup(set(A))) ).

% Union_natural
tff(fact_3700_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F3: fun(A,nat),X: A] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(A,nat,F3,X)),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(A,nat,F3,X))) ) ).

% funpow_times_power
tff(fact_3701_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,N2: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N2)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3)) ).

% funpow.simps(2)
tff(fact_3702_funpow__Suc__right,axiom,
    ! [A: $tType,N2: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N2)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3)),F3) ).

% funpow_Suc_right
tff(fact_3703_in__these__eq,axiom,
    ! [A: $tType,X: A,A5: set(option(A))] :
      ( pp(aa(set(A),bool,member(A,X),these(A,A5)))
    <=> pp(aa(set(option(A)),bool,member(option(A),aa(A,option(A),some(A),X)),A5)) ) ).

% in_these_eq
tff(fact_3704_funpow__add,axiom,
    ! [A: $tType,M: nat,N2: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3)) ).

% funpow_add
tff(fact_3705_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344872417868541ns_and(int,X,Y))) ) ).

% AND_lower
tff(fact_3706_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,X,Y)),X)) ) ).

% AND_upper1
tff(fact_3707_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,X,Y)),Y)) ) ).

% AND_upper2
tff(fact_3708_AND__upper1_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,Y,Ya)),Z2)) ) ) ).

% AND_upper1'
tff(fact_3709_AND__upper2_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,X,Y)),Z2)) ) ) ).

% AND_upper2'
tff(fact_3710_mod__star__trueI,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H))
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H)) ) ).

% mod_star_trueI
tff(fact_3711_mod__star__trueE,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H))
     => ~ ! [H5: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H5)) ) ).

% mod_star_trueE
tff(fact_3712_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_3713_top__assn__def,axiom,
    top_top(assn) = abs_assn(in_range) ).

% top_assn_def
tff(fact_3714_SUP__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: fun(bool,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(bool),set(A),image2(bool,A,A5),top_top(set(bool)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(bool,A,A5,fTrue)),aa(bool,A,A5,fFalse)) ) ).

% SUP_UNIV_bool_expand
tff(fact_3715_relpowp__fun__conv,axiom,
    ! [A: $tType,N2: nat,P: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N2),P),X),Y))
    <=> ? [F10: fun(nat,A)] :
          ( ( aa(nat,A,F10,zero_zero(nat)) = X )
          & ( aa(nat,A,F10,N2) = Y )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,F10,I4)),aa(nat,A,F10,aa(nat,nat,suc,I4)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_3716_INF__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: fun(bool,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(bool),set(A),image2(bool,A,A5),top_top(set(bool)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(bool,A,A5,fTrue)),aa(bool,A,A5,fFalse)) ) ).

% INF_UNIV_bool_expand
tff(fact_3717_Un__eq__UN,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(bool),set(set(A)),image2(bool,set(A),aa(set(A),fun(bool,set(A)),aTP_Lamp_pt(set(A),fun(set(A),fun(bool,set(A))),A5),B4)),top_top(set(bool)))) ).

% Un_eq_UN
tff(fact_3718_UN__bool__eq,axiom,
    ! [A: $tType,A5: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(bool),set(set(A)),image2(bool,set(A),A5),top_top(set(bool)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(bool,set(A),A5,fTrue)),aa(bool,set(A),A5,fFalse)) ).

% UN_bool_eq
tff(fact_3719_INT__bool__eq,axiom,
    ! [A: $tType,A5: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(bool),set(set(A)),image2(bool,set(A),A5),top_top(set(bool)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(bool,set(A),A5,fTrue)),aa(bool,set(A),A5,fFalse)) ).

% INT_bool_eq
tff(fact_3720_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,K,L)),K)) ) ).

% and_less_eq
tff(fact_3721_AND__upper1_H_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344872417868541ns_and(int,Y,Ya)),Z2)) ) ) ).

% AND_upper1''
tff(fact_3722_AND__upper2_H_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344872417868541ns_and(int,X,Y)),Z2)) ) ) ).

% AND_upper2''
tff(fact_3723_relpowp__bot,axiom,
    ! [A: $tType,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N2),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).

% relpowp_bot
tff(fact_3724_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H),bot_bot(set(nat))))) ).

% mod_h_bot_iff(2)
tff(fact_3725_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N2: nat] : aa(nat,A,semiring_1_of_nat(A),N2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_3726_and__not__num_Osimps_I2_J,axiom,
    ! [N2: num] : bit_and_not_num(one2,aa(num,num,bit0,N2)) = aa(num,option(num),some(num),one2) ).

% and_not_num.simps(2)
tff(fact_3727_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit0,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% and_not_num.simps(4)
tff(fact_3728_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_ri4277139882892585799ns_not(A,bit_se2239418461657761734s_mask(A,N2))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_3729_prod_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C4: set(set(B)),G3: fun(B,A)] :
          ( ! [X3: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X3),C4))
             => pp(aa(set(B),bool,finite_finite2(B),X3)) )
         => ( ! [X3: set(B)] :
                ( pp(aa(set(set(B)),bool,member(set(B),X3),C4))
               => ! [Xa4: set(B)] :
                    ( pp(aa(set(set(B)),bool,member(set(B),Xa4),C4))
                   => ( ( X3 != Xa4 )
                     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X3),Xa4) = bot_bot(set(B)) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C4)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G3),C4) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_3730_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_3731_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_3732_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% and_not_num.simps(7)
tff(fact_3733_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_pu(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_3734_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_pu(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_3735_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_pu(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_3736_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_pu(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N2)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_3737_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_3738_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_3739_sum_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C4: set(set(B)),G3: fun(B,A)] :
          ( ! [X3: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X3),C4))
             => pp(aa(set(B),bool,finite_finite2(B),X3)) )
         => ( ! [X3: set(B)] :
                ( pp(aa(set(set(B)),bool,member(set(B),X3),C4))
               => ! [Xa4: set(B)] :
                    ( pp(aa(set(set(B)),bool,member(set(B),Xa4),C4))
                   => ( ( X3 != Xa4 )
                     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X3),Xa4) = bot_bot(set(B)) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C4)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G3),C4) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_3740_Some__image__these__eq,axiom,
    ! [A: $tType,A5: set(option(A))] : aa(set(A),set(option(A)),image2(A,option(A),some(A)),these(A,A5)) = aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_pv(set(option(A)),fun(option(A),bool),A5)) ).

% Some_image_these_eq
tff(fact_3741_execute__ureturn,axiom,
    ! [A: $tType,X: A] : heap_Time_execute(A,heap_Time_ureturn(A,X)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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_pw(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_ureturn
tff(fact_3742_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( bit_se5824344872417868541ns_and(int,X,Xa2) = Y )
     => ( ( ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))) ) )
        & ( ~ ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% and_int.elims
tff(fact_3743_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa(set(int),bool,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
          & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( bit_se5824344872417868541ns_and(int,K,L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))) ) )
      & ( ~ ( pp(aa(set(int),bool,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( bit_se5824344872417868541ns_and(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.simps
tff(fact_3744_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( bit_se5824344872417868541ns_and(int,X,Xa2) = Y )
     => ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)))
       => ~ ( ( ( ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                  & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))) ) )
              & ( ~ ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))) ) ) ) ).

% and_int.pelims
tff(fact_3745_insertCI,axiom,
    ! [A: $tType,A3: A,B4: set(A),B2: A] :
      ( ( ~ pp(aa(set(A),bool,member(A,A3),B4))
       => ( A3 = B2 ) )
     => pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B4))) ) ).

% insertCI
tff(fact_3746_insert__iff,axiom,
    ! [A: $tType,A3: A,B2: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A5)))
    <=> ( ( A3 = B2 )
        | pp(aa(set(A),bool,member(A,A3),A5)) ) ) ).

% insert_iff
tff(fact_3747_insert__absorb2,axiom,
    ! [A: $tType,X: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5) ).

% insert_absorb2
tff(fact_3748_top1I,axiom,
    ! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).

% top1I
tff(fact_3749_image__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: B,B4: set(B)] : aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),B4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F3,A3)),aa(set(B),set(A),image2(B,A,F3),B4)) ).

% image_insert
tff(fact_3750_insert__image,axiom,
    ! [B: $tType,A: $tType,X: A,A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,X)),aa(set(A),set(B),image2(A,B,F3),A5)) = aa(set(A),set(B),image2(A,B,F3),A5) ) ) ).

% insert_image
tff(fact_3751_singletonI,axiom,
    ! [A: $tType,A3: A] : pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) ).

% singletonI
tff(fact_3752_insert__subset,axiom,
    ! [A: $tType,X: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),B4))
    <=> ( pp(aa(set(A),bool,member(A,X),B4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ) ).

% insert_subset
tff(fact_3753_Int__insert__right__if1,axiom,
    ! [A: $tType,A3: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_3754_Int__insert__right__if0,axiom,
    ! [A: $tType,A3: A,A5: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),A5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ) ) ).

% Int_insert_right_if0
tff(fact_3755_insert__inter__insert,axiom,
    ! [A: $tType,A3: A,A5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ).

% insert_inter_insert
tff(fact_3756_Int__insert__left__if1,axiom,
    ! [A: $tType,A3: A,C4: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),C4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) ) ) ).

% Int_insert_left_if1
tff(fact_3757_Int__insert__left__if0,axiom,
    ! [A: $tType,A3: A,C4: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),C4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4) ) ) ).

% Int_insert_left_if0
tff(fact_3758_Un__insert__right,axiom,
    ! [A: $tType,A5: set(A),A3: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) ).

% Un_insert_right
tff(fact_3759_Un__insert__left,axiom,
    ! [A: $tType,A3: A,B4: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C4)) ).

% Un_insert_left
tff(fact_3760_Diff__insert0,axiom,
    ! [A: $tType,X: A,A5: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) ) ) ).

% Diff_insert0
tff(fact_3761_insert__Diff1,axiom,
    ! [A: $tType,X: A,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),B4))
     => ( 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)),insert(A),X),A5)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) ) ) ).

% insert_Diff1
tff(fact_3762_top2I,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X),Y)) ).

% top2I
tff(fact_3763_singleton__conv,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_br(A,fun(A,bool),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ).

% singleton_conv
tff(fact_3764_singleton__conv2,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),fequal(A),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_3765_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A3: A,A5: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5) )
    <=> ( ( A3 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq
tff(fact_3766_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A3: A,A5: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
    <=> ( ( A3 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_3767_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% cSup_singleton
tff(fact_3768_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% cInf_singleton
tff(fact_3769_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A3: A,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)),B4) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),B4))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_3770_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A3: A,A5: set(A),B4: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)),B4) )
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),B4))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_3771_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B4: set(A),A3: A,A5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),B4))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),A5) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_3772_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A5: set(A),B2: A,B4: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B4)) )
    <=> ( ~ pp(aa(set(A),bool,member(A,B2),A5))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_3773_Sup__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ).

% Sup_insert
tff(fact_3774_insert__Diff__single,axiom,
    ! [A: $tType,A3: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5) ).

% insert_Diff_single
tff(fact_3775_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),bot_bot(set(A))) )
        <=> ( ( A3 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3776_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)),insert(A),A3),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_3777_Inf__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A5)) ) ).

% Inf_insert
tff(fact_3778_these__insert__Some,axiom,
    ! [A: $tType,X: A,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),aa(A,option(A),some(A),X)),A5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),these(A,A5)) ).

% these_insert_Some
tff(fact_3779_prod_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ~ pp(aa(set(B),bool,member(B,X),A5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)) ) ) ) ) ).

% prod.insert
tff(fact_3780_subset__Compl__singleton,axiom,
    ! [A: $tType,A5: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
    <=> ~ pp(aa(set(A),bool,member(A,B2),A5)) ) ).

% subset_Compl_singleton
tff(fact_3781_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_3782_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_px(A,fun(B,A)),X)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ).

% range_constant
tff(fact_3783_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C4: set(B),A3: A,B4: fun(B,set(A))] :
      ( ( ( C4 = bot_bot(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_py(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C4)) = bot_bot(set(A)) ) )
      & ( ( C4 != bot_bot(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_py(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C4))) ) ) ) ).

% UN_simps(1)
tff(fact_3784_UN__singleton,axiom,
    ! [A: $tType,A5: 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_pz(A,set(A))),A5)) = A5 ).

% UN_singleton
tff(fact_3785_UN__insert,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: B,A5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B4,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))) ).

% UN_insert
tff(fact_3786_INT__insert,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: B,A5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B4,A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))) ).

% INT_insert
tff(fact_3787_map__update__eta__repair_I2_J,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V2: B] :
      ( ( aa(A,option(B),M,K) = none(B) )
     => ( ran(A,B,aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_qa(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),M),K),V2)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),V2),ran(A,B,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_3788_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ( A3 != B2 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_3789_insert__UNIV,axiom,
    ! [A: $tType,X: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_3790_singleton__inject,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
     => ( A3 = B2 ) ) ).

% singleton_inject
tff(fact_3791_insert__not__empty,axiom,
    ! [A: $tType,A3: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_3792_doubleton__eq__iff,axiom,
    ! [A: $tType,A3: A,B2: A,C2: A,D3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),D3),bot_bot(set(A)))) )
    <=> ( ( ( A3 = C2 )
          & ( B2 = D3 ) )
        | ( ( A3 = D3 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_3793_singleton__iff,axiom,
    ! [A: $tType,B2: A,A3: A] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))
    <=> ( B2 = A3 ) ) ).

% singleton_iff
tff(fact_3794_singletonD,axiom,
    ! [A: $tType,B2: A,A3: A] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))
     => ( B2 = A3 ) ) ).

% singletonD
tff(fact_3795_Int__insert__right,axiom,
    ! [A: $tType,A3: A,A5: set(A),B4: set(A)] :
      ( ( pp(aa(set(A),bool,member(A,A3),A5))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,A3),A5))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ) ) ) ).

% Int_insert_right
tff(fact_3796_Int__insert__left,axiom,
    ! [A: $tType,A3: A,C4: set(A),B4: set(A)] :
      ( ( pp(aa(set(A),bool,member(A,A3),C4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,A3),C4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C4) ) ) ) ).

% Int_insert_left
tff(fact_3797_less__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F5: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less(filter(A)),F4),F5))
    <=> ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F5))
        & ~ pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),F4)) ) ) ).

% less_filter_def
tff(fact_3798_insert__compr,axiom,
    ! [A: $tType,A3: A,B4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_qb(A,fun(set(A),fun(A,bool)),A3),B4)) ).

% insert_compr
tff(fact_3799_insert__Collect,axiom,
    ! [A: $tType,A3: A,P: fun(A,bool)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(fun(A,bool),set(A),collect(A),P)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_qc(A,fun(fun(A,bool),fun(A,bool)),A3),P)) ).

% insert_Collect
tff(fact_3800_insert__mono,axiom,
    ! [A: $tType,C4: set(A),D4: set(A),A3: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),D4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),C4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),D4))) ) ).

% insert_mono
tff(fact_3801_subset__insert,axiom,
    ! [A: $tType,X: A,A5: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B4)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ) ).

% subset_insert
tff(fact_3802_subset__insertI,axiom,
    ! [A: $tType,B4: set(A),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4))) ).

% subset_insertI
tff(fact_3803_subset__insertI2,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B4))) ) ).

% subset_insertI2
tff(fact_3804_insert__subsetI,axiom,
    ! [A: $tType,X: A,A5: set(A),X6: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),X6)),A5)) ) ) ).

% insert_subsetI
tff(fact_3805_insert__Diff__if,axiom,
    ! [A: $tType,X: A,B4: set(A),A5: set(A)] :
      ( ( pp(aa(set(A),bool,member(A,X),B4))
       => ( 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)),insert(A),X),A5)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),B4))
       => ( 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)),insert(A),X),A5)),B4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) ) ) ) ).

% insert_Diff_if
tff(fact_3806_insertE,axiom,
    ! [A: $tType,A3: A,B2: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A5)))
     => ( ( A3 != B2 )
       => pp(aa(set(A),bool,member(A,A3),A5)) ) ) ).

% insertE
tff(fact_3807_insertI1,axiom,
    ! [A: $tType,A3: A,B4: set(A)] : pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4))) ).

% insertI1
tff(fact_3808_insertI2,axiom,
    ! [A: $tType,A3: A,B4: set(A),B2: A] :
      ( pp(aa(set(A),bool,member(A,A3),B4))
     => pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B4))) ) ).

% insertI2
tff(fact_3809_Set_Oset__insert,axiom,
    ! [A: $tType,X: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ~ ! [B10: set(A)] :
            ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B10) )
           => pp(aa(set(A),bool,member(A,X),B10)) ) ) ).

% Set.set_insert
tff(fact_3810_insert__ident,axiom,
    ! [A: $tType,X: A,A5: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A5))
     => ( ~ pp(aa(set(A),bool,member(A,X),B4))
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B4) )
        <=> ( A5 = B4 ) ) ) ) ).

% insert_ident
tff(fact_3811_insert__absorb,axiom,
    ! [A: $tType,A3: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5) = A5 ) ) ).

% insert_absorb
tff(fact_3812_insert__eq__iff,axiom,
    ! [A: $tType,A3: A,A5: set(A),B2: A,B4: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),A5))
     => ( ~ pp(aa(set(A),bool,member(A,B2),B4))
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B4) )
        <=> ( ( ( A3 = B2 )
             => ( A5 = B4 ) )
            & ( ( A3 != B2 )
             => ? [C8: set(A)] :
                  ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),C8) )
                  & ~ pp(aa(set(A),bool,member(A,B2),C8))
                  & ( B4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),C8) )
                  & ~ pp(aa(set(A),bool,member(A,A3),C8)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_3813_insert__commute,axiom,
    ! [A: $tType,X: A,Y: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),A5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) ).

% insert_commute
tff(fact_3814_mk__disjoint__insert,axiom,
    ! [A: $tType,A3: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ? [B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B10) )
          & ~ pp(aa(set(A),bool,member(A,A3),B10)) ) ) ).

% mk_disjoint_insert
tff(fact_3815_Collect__conv__if,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( ( pp(aa(A,bool,P,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_qd(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_qd(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if
tff(fact_3816_Collect__conv__if2,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( ( pp(aa(A,bool,P,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_qe(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_qe(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if2
tff(fact_3817_insert__def,axiom,
    ! [A: $tType,A3: A,B4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_br(A,fun(A,bool),A3))),B4) ).

% insert_def
tff(fact_3818_finite_Ocases,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( A3 != bot_bot(set(A)) )
       => ~ ! [A11: set(A)] :
              ( ? [A6: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A6),A11)
             => ~ pp(aa(set(A),bool,finite_finite2(A),A11)) ) ) ) ).

% finite.cases
tff(fact_3819_finite_Osimps,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ? [A13: set(A),A8: A] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A8),A13) )
            & pp(aa(set(A),bool,finite_finite2(A),A13)) ) ) ) ).

% finite.simps
tff(fact_3820_finite__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [X3: A,F8: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),F8))
             => ( ~ pp(aa(set(A),bool,member(A,X3),F8))
               => ( pp(aa(set(A),bool,P,F8))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),F8))) ) ) )
         => pp(aa(set(A),bool,P,F4)) ) ) ) ).

% finite_induct
tff(fact_3821_finite__ne__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( ( F4 != bot_bot(set(A)) )
       => ( ! [X3: A] : pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A)))))
         => ( ! [X3: A,F8: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F8))
               => ( ( F8 != bot_bot(set(A)) )
                 => ( ~ pp(aa(set(A),bool,member(A,X3),F8))
                   => ( pp(aa(set(A),bool,P,F8))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),F8))) ) ) ) )
           => pp(aa(set(A),bool,P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3822_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),A5: set(A)] :
      ( ! [A11: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A11))
         => pp(aa(set(A),bool,P,A11)) )
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [X3: A,F8: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),F8))
             => ( ~ pp(aa(set(A),bool,member(A,X3),F8))
               => ( pp(aa(set(A),bool,P,F8))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),F8))) ) ) )
         => pp(aa(set(A),bool,P,A5)) ) ) ) ).

% infinite_finite_induct
tff(fact_3823_subset__singletonD,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))
     => ( ( A5 = bot_bot(set(A)) )
        | ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_3824_subset__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),A3: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))
    <=> ( ( X6 = bot_bot(set(A)) )
        | ( X6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_3825_Sup__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ).

% Sup_finite_insert
tff(fact_3826_singleton__Un__iff,axiom,
    ! [A: $tType,X: A,A5: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) )
    <=> ( ( ( A5 = bot_bot(set(A)) )
          & ( B4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B4 = bot_bot(set(A)) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_3827_Un__singleton__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),X: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
    <=> ( ( ( A5 = bot_bot(set(A)) )
          & ( B4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B4 = bot_bot(set(A)) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_3828_insert__is__Un,axiom,
    ! [A: $tType,A3: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))),A5) ).

% insert_is_Un
tff(fact_3829_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A5))
     => ( 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)),insert(A),X),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = A5 ) ) ).

% Diff_insert_absorb
tff(fact_3830_Diff__insert2,axiom,
    ! [A: $tType,A5: set(A),A3: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),B4) ).

% Diff_insert2
tff(fact_3831_insert__Diff,axiom,
    ! [A: $tType,A3: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = A5 ) ) ).

% insert_Diff
tff(fact_3832_Diff__insert,axiom,
    ! [A: $tType,A5: set(A),A3: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B4)) = 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)),A5),B4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_3833_set__minus__singleton__eq,axiom,
    ! [A: $tType,X: A,X6: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),X6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_3834_insert__minus__eq,axiom,
    ! [A: $tType,X: A,Y: A,A5: set(A)] :
      ( ( X != Y )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_3835_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
         => ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3836_Inf__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A5)) ) ).

% Inf_finite_insert
tff(fact_3837_subset__Diff__insert,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),X: A,C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),C4))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C4)))
        & ~ pp(aa(set(A),bool,member(A,X),A5)) ) ) ).

% subset_Diff_insert
tff(fact_3838_card__insert__le,axiom,
    ! [A: $tType,A5: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)))) ).

% card_insert_le
tff(fact_3839_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,A5: set(B),C2: A] :
      ( ( ( A5 = bot_bot(set(B)) )
       => ( aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_px(A,fun(B,A)),C2)),A5) = bot_bot(set(A)) ) )
      & ( ( A5 != bot_bot(set(B)) )
       => ( aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_px(A,fun(B,A)),C2)),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),bot_bot(set(A))) ) ) ) ).

% image_constant_conv
tff(fact_3840_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A5: set(A),C2: B] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_qf(B,fun(A,B),C2)),A5) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_3841_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A5: set(A),A3: B,B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,U),A5))
     => ( 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_qg(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B4)),A5)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5))) ) ) ).

% UN_insert_distrib
tff(fact_3842_INT__extend__simps_I5_J,axiom,
    ! [I6: $tType,J4: $tType,A3: I6,B4: fun(J4,set(I6)),C4: set(J4)] : aa(set(I6),set(I6),aa(I6,fun(set(I6),set(I6)),insert(I6),A3),aa(set(set(I6)),set(I6),complete_Inf_Inf(set(I6)),aa(set(J4),set(set(I6)),image2(J4,set(I6),B4),C4))) = aa(set(set(I6)),set(I6),complete_Inf_Inf(set(I6)),aa(set(J4),set(set(I6)),image2(J4,set(I6),aa(fun(J4,set(I6)),fun(J4,set(I6)),aTP_Lamp_qh(I6,fun(fun(J4,set(I6)),fun(J4,set(I6))),A3),B4)),C4)) ).

% INT_extend_simps(5)
tff(fact_3843_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A5: set(A),A3: B,B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,U),A5))
     => ( 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_qg(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B4)),A5)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5))) ) ) ).

% INT_insert_distrib
tff(fact_3844_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [S: set(B),P: fun(set(B),bool),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(B),bool,P,bot_bot(set(B))))
           => ( ! [X3: B,S8: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite2(B),S8))
                 => ( ! [Y4: B] :
                        ( pp(aa(set(B),bool,member(B,Y4),S8))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,Y4)),aa(B,A,F3,X3))) )
                   => ( pp(aa(set(B),bool,P,S8))
                     => pp(aa(set(B),bool,P,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X3),S8))) ) ) )
             => pp(aa(set(B),bool,P,S)) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3845_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B5: A,A11: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),A11))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),B5)) )
                   => ( pp(aa(set(A),bool,P,A11))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B5),A11))) ) ) )
             => pp(aa(set(A),bool,P,A5)) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3846_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B5: A,A11: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),A11))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B5),X5)) )
                   => ( pp(aa(set(A),bool,P,A11))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B5),A11))) ) ) )
             => pp(aa(set(A),bool,P,A5)) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3847_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A5: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F4),A5))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A6: A,F8: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F8))
               => ( pp(aa(set(A),bool,member(A,A6),A5))
                 => ( ~ pp(aa(set(A),bool,member(A,A6),F8))
                   => ( pp(aa(set(A),bool,P,F8))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A6),F8))) ) ) ) )
           => pp(aa(set(A),bool,P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3848_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A5: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F4),A5))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A6: A,F8: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F8))
               => ( pp(aa(set(A),bool,member(A,A6),A5))
                 => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F8),A5))
                   => ( ~ pp(aa(set(A),bool,member(A,A6),F8))
                     => ( pp(aa(set(A),bool,P,F8))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A6),F8))) ) ) ) ) )
           => pp(aa(set(A),bool,P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3849_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: A,X: B] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) )
     => ( aa(B,A,F3,X) = A3 ) ) ).

% range_eq_singletonD
tff(fact_3850_infinite__remove,axiom,
    ! [A: $tType,S: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
     => ~ pp(aa(set(A),bool,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)),insert(A),A3),bot_bot(set(A)))))) ) ).

% infinite_remove
tff(fact_3851_infinite__coinduct,axiom,
    ! [A: $tType,X6: fun(set(A),bool),A5: set(A)] :
      ( pp(aa(set(A),bool,X6,A5))
     => ( ! [A11: set(A)] :
            ( pp(aa(set(A),bool,X6,A11))
           => ? [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),A11))
                & ( pp(aa(set(A),bool,X6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A11),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A))))))
                  | ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A11),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A)))))) ) ) )
       => ~ pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% infinite_coinduct
tff(fact_3852_finite__empty__induct,axiom,
    ! [A: $tType,A5: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,P,A5))
       => ( ! [A6: A,A11: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A11))
             => ( pp(aa(set(A),bool,member(A,A6),A11))
               => ( pp(aa(set(A),bool,P,A11))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A11),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A6),bot_bot(set(A)))))) ) ) )
         => pp(aa(set(A),bool,P,bot_bot(set(A)))) ) ) ) ).

% finite_empty_induct
tff(fact_3853_prod_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( pp(aa(set(B),bool,member(B,X),A5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) ) )
            & ( ~ pp(aa(set(B),bool,member(B,X),A5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)) ) ) ) ) ) ).

% prod.insert_if
tff(fact_3854_Diff__single__insert,axiom,
    ! [A: $tType,A5: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B4))) ) ).

% Diff_single_insert
tff(fact_3855_subset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B4)))
    <=> ( ( pp(aa(set(A),bool,member(A,X),A5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B4)) )
        & ( ~ pp(aa(set(A),bool,member(A,X),A5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_3856_card__1__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) )
     => ~ ! [X3: A] : A5 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_3857_remove__subset,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),S))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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)),insert(A),X),bot_bot(set(A))))),S)) ) ).

% remove_subset
tff(fact_3858_Union__image__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),A3: B,B4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),B4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F3,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),B4))) ).

% Union_image_insert
tff(fact_3859_Compl__insert,axiom,
    ! [A: $tType,X: A,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = 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)),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_3860_in__image__insert__iff,axiom,
    ! [A: $tType,B4: set(set(A)),X: A,A5: set(A)] :
      ( ! [C6: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),C6),B4))
         => ~ pp(aa(set(A),bool,member(A,X),C6)) )
     => ( pp(aa(set(set(A)),bool,member(set(A),A5),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X)),B4)))
      <=> ( pp(aa(set(A),bool,member(A,X),A5))
          & pp(aa(set(set(A)),bool,member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B4)) ) ) ) ).

% in_image_insert_iff
tff(fact_3861_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A3: B,A5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A5))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F3,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))) ) ).

% SUP_insert
tff(fact_3862_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A3: B,A5: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A5))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F3,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))) ) ).

% INF_insert
tff(fact_3863_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C4: set(B),A3: A,B4: fun(B,set(A))] :
      ( ( ( C4 = bot_bot(set(B)) )
       => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C4))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) )
      & ( ( C4 != bot_bot(set(B)) )
       => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C4))) = 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_py(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C4)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_3864_finite__remove__induct,axiom,
    ! [A: $tType,B4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [A11: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A11))
             => ( ( A11 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),B4))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A11),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A11)) ) ) ) )
         => pp(aa(set(A),bool,P,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_3865_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),B4: set(A)] :
      ( pp(aa(set(A),bool,P,bot_bot(set(A))))
     => ( ( ~ pp(aa(set(A),bool,finite_finite2(A),B4))
         => pp(aa(set(A),bool,P,B4)) )
       => ( ! [A11: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A11))
             => ( ( A11 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),B4))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A11),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A11)) ) ) ) )
         => pp(aa(set(A),bool,P,B4)) ) ) ) ).

% remove_induct
tff(fact_3866_card__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
    <=> ? [B13: A,B7: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B13),B7) )
          & ~ pp(aa(set(A),bool,member(A,B13),B7))
          & ( aa(set(A),nat,finite_card(A),B7) = K )
          & ( ( K = zero_zero(nat) )
           => ( B7 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3867_card__eq__SucD,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
     => ? [B5: A,B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B5),B10) )
          & ~ pp(aa(set(A),bool,member(A,B5),B10))
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3868_card__1__singleton__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X4: A] : A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_3869_card__le__Suc__iff,axiom,
    ! [A: $tType,N2: nat,A5: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),aa(set(A),nat,finite_card(A),A5)))
    <=> ? [A8: A,B7: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A8),B7) )
          & ~ pp(aa(set(A),bool,member(A,A8),B7))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(set(A),nat,finite_card(A),B7)))
          & pp(aa(set(A),bool,finite_finite2(A),B7)) ) ) ).

% card_le_Suc_iff
tff(fact_3870_card__1__singletonI,axiom,
    ! [A: $tType,S: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( ( aa(set(A),nat,finite_card(A),S) = one_one(nat) )
       => ( pp(aa(set(A),bool,member(A,X),S))
         => ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_3871_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,K: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).

% Iio_Int_singleton
tff(fact_3872_card__Diff1__le,axiom,
    ! [A: $tType,A5: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))) ).

% card_Diff1_le
tff(fact_3873_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [T7: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T7),S))
             => ( pp(aa(set(A),bool,P,T7))
               => ? [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T7)))
                    & pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),T7))) ) ) )
         => pp(aa(set(A),bool,P,S)) ) ) ) ).

% finite_induct_select
tff(fact_3874_psubset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B4)))
    <=> ( ( pp(aa(set(A),bool,member(A,X),B4))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4)) )
        & ( ~ pp(aa(set(A),bool,member(A,X),B4))
         => ( ( pp(aa(set(A),bool,member(A,X),A5))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B4)) )
            & ( ~ pp(aa(set(A),bool,member(A,X),A5))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3875_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] : set_or7035219750837199246ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3876_sum__diff1__nat,axiom,
    ! [A: $tType,A3: A,A5: set(A),F3: fun(A,nat)] :
      ( ( pp(aa(set(A),bool,member(A,A3),A5))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(A,nat,F3,A3)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,A3),A5))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) ) ) ) ).

% sum_diff1_nat
tff(fact_3877_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_3878_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N2)))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N2)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N2)),set_or1337092689740270186AtMost(int,M,N2)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3879_simp__from__to,axiom,
    ! [J: int,I2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( set_or1337092689740270186AtMost(int,I2,J) = bot_bot(set(int)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( set_or1337092689740270186AtMost(int,I2,J) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),I2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ) ).

% simp_from_to
tff(fact_3880_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: 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_qi(fun(B,A),fun(B,set(A)),F3)),A5)) = aa(set(B),set(A),image2(B,A,F3),A5) ).

% UNION_singleton_eq_range
tff(fact_3881_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G3: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3882_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,member(B,X),A5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_3883_sum__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A5: set(B),A3: B,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( pp(aa(set(B),bool,member(B,A3),A5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))) = 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),F3),A5)),aa(B,A,F3,A3)) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),A5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5) ) ) ) ) ) ).

% sum_diff1
tff(fact_3884_card__2__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X4: A,Y5: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),bot_bot(set(A)))) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3885_prod_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G3: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3886_prod_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,member(B,X),A5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% prod.remove
tff(fact_3887_card_Oremove,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_3888_card_Oinsert__remove,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_3889_card__Suc__Diff1,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_Suc_Diff1
tff(fact_3890_card__insert__disjoint_H,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ~ pp(aa(set(A),bool,member(A,X),A5))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5))),aa(nat,nat,suc,zero_zero(nat))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_insert_disjoint'
tff(fact_3891_card__Diff1__less,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))) ) ) ).

% card_Diff1_less
tff(fact_3892_card__Diff2__less,axiom,
    ! [A: $tType,A5: set(A),X: A,Y: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( pp(aa(set(A),bool,member(A,Y),A5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))) ) ) ) ).

% card_Diff2_less
tff(fact_3893_card__Diff1__less__iff,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
        & pp(aa(set(A),bool,member(A,X),A5)) ) ) ).

% card_Diff1_less_iff
tff(fact_3894_card__3__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X4: A,Y5: A,Z6: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z6),bot_bot(set(A))))) )
          & ( X4 != Y5 )
          & ( Y5 != Z6 )
          & ( X4 != Z6 ) ) ) ).

% card_3_iff
tff(fact_3895_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_3896_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3897_card__Diff__singleton__if,axiom,
    ! [A: $tType,X: A,A5: set(A)] :
      ( ( pp(aa(set(A),bool,member(A,X),A5))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),A5))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_Diff_singleton_if
tff(fact_3898_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qj(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),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)),insert(B),A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qj(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),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)),insert(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_3899_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qk(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),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)),insert(B),A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qk(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),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)),insert(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_3900_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I2: C,A5: set(C),F3: fun(C,B)] :
          ( pp(aa(set(C),bool,member(C,I2),A5))
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,member(C,X3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A5),aa(set(C),set(C),aa(C,fun(set(C),set(C)),insert(C),I2),bot_bot(set(C))))))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F3,X3))) )
           => ( pp(aa(set(C),bool,finite_finite2(C),A5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F3,I2)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F3),A5))) ) ) ) ) ).

% member_le_sum
tff(fact_3901_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A5: set(B),F3: fun(B,A),A3: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( aa(B,A,F3,A3) != zero_zero(A) )
           => ( ( pp(aa(set(B),bool,member(B,A3),A5))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)),aa(B,A,F3,A3)) ) )
              & ( ~ pp(aa(set(B),bool,member(B,A3),A5))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_3902_card__insert__le__m1,axiom,
    ! [A: $tType,N2: nat,Y: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),Y))),N2)) ) ) ).

% card_insert_le_m1
tff(fact_3903_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
     => ( ! [K2: int,L4: int] :
            ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L4)))
           => ( ( ~ ( pp(aa(set(int),bool,member(int,K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,member(int,L4),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L4),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,K2),L4)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_3904_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L)))
     => ( ( ( pp(aa(set(int),bool,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( bit_se5824344872417868541ns_and(int,K,L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,member(int,K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( bit_se5824344872417868541ns_and(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_3905_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% ccpo_Sup_singleton
tff(fact_3906_execute__return,axiom,
    ! [A: $tType,X: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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_ql(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_return
tff(fact_3907_ureturn__def,axiom,
    ! [A: $tType,X: A] : heap_Time_ureturn(A,X) = heap_Time_heap(A,aTP_Lamp_pw(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% ureturn_def
tff(fact_3908_the__elem__eq,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ).

% the_elem_eq
tff(fact_3909_subset__mset_OcInf__singleton,axiom,
    ! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.cInf_singleton
tff(fact_3910_subset__mset_OcSup__singleton,axiom,
    ! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.cSup_singleton
tff(fact_3911_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_3912_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3913_these__insert__None,axiom,
    ! [A: $tType,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),A5)) = these(A,A5) ).

% these_insert_None
tff(fact_3914_singleton__None__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A))))) = none(A) ) ) ).

% singleton_None_Sup
tff(fact_3915_UNIV__bool,axiom,
    top_top(set(bool)) = aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fFalse),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))) ).

% UNIV_bool
tff(fact_3916_Union__insert,axiom,
    ! [A: $tType,A3: set(A),B4: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) ).

% Union_insert
tff(fact_3917_Inter__insert,axiom,
    ! [A: $tType,A3: set(A),B4: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4)) ).

% Inter_insert
tff(fact_3918_insert__partition,axiom,
    ! [A: $tType,X: set(A),F4: set(set(A))] :
      ( ~ pp(aa(set(set(A)),bool,member(set(A),X),F4))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),X),F4)))
           => ! [Xa4: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),Xa4),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),X),F4)))
               => ( ( X3 != Xa4 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_3919_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N2)) = set_or1337092689740270186AtMost(nat,M,N2) ) ) ).

% atLeastAtMost_insertL
tff(fact_3920_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N2)))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,N2)),set_or1337092689740270186AtMost(nat,M,N2)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3921_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( set_or1337092689740270186AtMost(nat,M,N2) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N2)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3922_return__def,axiom,
    ! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),X) = heap_Time_heap(A,aTP_Lamp_ql(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% return_def
tff(fact_3923_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N2),set_or7035219750837199246ssThan(nat,M,N2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_3924_these__not__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) != bot_bot(set(A)) )
    <=> ( ( B4 != bot_bot(set(option(A))) )
        & ( B4 != aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_3925_these__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) = bot_bot(set(A)) )
    <=> ( ( B4 = bot_bot(set(option(A))) )
        | ( B4 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_3926_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B),X: A] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( pp(aa(set(A),bool,member(A,Y3),A5))
           => ( aa(A,B,F3,Y3) = aa(A,B,F3,X) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F3),A5)) = aa(A,B,F3,X) ) ) ) ).

% the_elem_image_unique
tff(fact_3927_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N2) = 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),N2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_3928_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N2) = 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),N2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3929_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_3930_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_3931_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
       => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_qm(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_qm(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_qm(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_3932_Sup__option__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(option(A))] :
          ( ( ( ( A5 = bot_bot(set(option(A))) )
              | ( A5 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) )
           => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A5) = none(A) ) )
          & ( ~ ( ( A5 = bot_bot(set(option(A))) )
                | ( A5 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) )
           => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A5) = aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),these(A,A5))) ) ) ) ) ).

% Sup_option_def
tff(fact_3933_execute__heap,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_execute(A,heap_Time_heap(A,F3)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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)))),F3) ).

% execute_heap
tff(fact_3934_heap__def,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_heap(A,F3) = heap_Time_Heap2(A,aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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)))),F3)) ).

% heap_def
tff(fact_3935_bind__return,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A)] : heap_Time_bind(A,A,F3,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_qn(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F3)) ).

% bind_return
tff(fact_3936_return__bind,axiom,
    ! [A: $tType,B: $tType,X: B,F3: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),F3) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_qo(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),X),F3)) ).

% return_bind
tff(fact_3937_bind__lift,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(B),G3: fun(B,A)] : heap_Time_bind(B,A,F3,heap_Time_lift(B,A,G3)) = heap_Time_bind(B,A,F3,aTP_Lamp_qp(fun(B,A),fun(B,heap_Time_Heap(A)),G3)) ).

% bind_lift
tff(fact_3938_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F3: fun(A,B),I2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),F4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_qq(set(A),fun(fun(A,B),fun(A,bool)),I5),F3))),F4))
           => ( ( pp(aa(set(A),bool,member(A,I2),I5))
               => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I5)),aa(A,B,F3,I2)) ) )
              & ( ~ pp(aa(set(A),bool,member(A,I2),I5))
               => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F3,I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3939_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P3,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_3940_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
         => ( ( pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1027152243600224163dd_sum(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = groups1027152243600224163dd_sum(B,A,P3,I5) ) )
            & ( ~ pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1027152243600224163dd_sum(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P3,I2)),groups1027152243600224163dd_sum(B,A,P3,I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_3941_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G3,aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_qr(fun(B,A),fun(set(B),fun(B,bool)),G3),I5))) = groups1027152243600224163dd_sum(B,A,G3,I5) ) ).

% sum.non_neutral'
tff(fact_3942_sum_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_du(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G3,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3943_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T2: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G3,T2) = groups1027152243600224163dd_sum(B,A,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3944_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T2: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,H,I3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G3,S) = groups1027152243600224163dd_sum(B,A,H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3945_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T2: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G3,T2) = groups1027152243600224163dd_sum(B,A,G3,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3946_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T2: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G3,S) = groups1027152243600224163dd_sum(B,A,G3,T2) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3947_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),G3))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_du(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G3,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3948_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P3: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1027152243600224163dd_sum(B,A,P3,I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1027152243600224163dd_sum(B,A,P3,I5) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_3949_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F3: fun(A,B),I2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_qq(set(A),fun(fun(A,B),fun(A,bool)),I5),F3))))
         => ( ( pp(aa(set(A),bool,member(A,I2),I5))
             => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I5)),aa(A,B,F3,I2)) ) )
            & ( ~ pp(aa(set(A),bool,member(A,I2),I5))
             => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F3,I5) ) ) ) ) ) ).

% sum_diff1'
tff(fact_3950_is__singleton__the__elem,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the_elem(A,A5)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_3951_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_3952_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),N2: nat,X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,N2,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),N2)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_3953_is__singletonI,axiom,
    ! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_3954_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( list_update(A,Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_3955_nth__list__update__eq,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_3956_nth__update__invalid,axiom,
    ! [A: $tType,I2: nat,L: list(A),J: nat,X: A] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( aa(nat,A,nth(A,list_update(A,L,J,X)),I2) = aa(nat,A,nth(A,L),I2) ) ) ).

% nth_update_invalid
tff(fact_3957_set__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_3958_distinct__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( distinct(A,list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_3959_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A),X: A,I2: nat] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A5))
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,X))),A5)) ) ) ).

% set_update_subsetI
tff(fact_3960_is__singletonI_H,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => ( pp(aa(set(A),bool,member(A,Y3),A5))
             => ( X3 = Y3 ) ) )
       => is_singleton(A,A5) ) ) ).

% is_singletonI'
tff(fact_3961_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,X))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),Xs)))) ).

% set_update_subset_insert
tff(fact_3962_in__set__upd__eq,axiom,
    ! [A: $tType,I2: nat,L: list(A),X: A,Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y))))
      <=> ( ( X = Y )
          | ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L)))
            & ! [Y5: A] : pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3963_in__set__upd__cases,axiom,
    ! [A: $tType,X: A,L: list(A),I2: nat,Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
         => ( X != Y ) )
       => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L))) ) ) ).

% in_set_upd_cases
tff(fact_3964_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I2: nat,L: list(A),X: A,Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y))))
      <=> ( ( X = Y )
          | ! [Y5: A] : pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3965_set__update__memI,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,Xs,N2,X)))) ) ).

% set_update_memI
tff(fact_3966_nth__list__update,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( ( I2 = J )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J) = X ) )
        & ( ( I2 != J )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_3967_list__update__same__conv,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( list_update(A,Xs,I2,X) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_3968_nth__list__update_H,axiom,
    ! [A: $tType,I2: nat,J: nat,L: list(A),X: A] :
      ( ( ( ( I2 = J )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))) )
       => ( aa(nat,A,nth(A,list_update(A,L,I2,X)),J) = X ) )
      & ( ~ ( ( I2 = J )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))) )
       => ( aa(nat,A,nth(A,list_update(A,L,I2,X)),J) = aa(nat,A,nth(A,L),J) ) ) ) ).

% nth_list_update'
tff(fact_3969_is__singleton__def,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ? [X4: A] : A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_3970_is__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
     => ~ ! [X3: A] : A5 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_3971_insert__swap__set__eq,axiom,
    ! [A: $tType,I2: nat,L: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,L),I2)),aa(list(A),set(A),set2(A),list_update(A,L,I2,X))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_3972_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A3: A,I2: nat] :
      ( distinct(A,Xs)
     => ( ~ pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),I2)),bot_bot(set(A))))))
       => distinct(A,list_update(A,Xs,I2,A3)) ) ) ).

% distinct_list_update
tff(fact_3973_lookup__chain,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(B)
     => ! [R3: ref(B),F3: heap_Time_Heap(A)] : heap_Time_bind(B,A,ref_lookup(B,R3),aTP_Lamp_qs(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),F3)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_qn(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F3)) ) ).

% lookup_chain
tff(fact_3974_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N5: num] : Xa2 = aa(num,num,bit0,N5)
             => ( Y != aa(num,option(num),some(num),one2) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N5: num] : Xa2 = aa(num,num,bit1,N5)
               => ( Y != none(num) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa(num,num,bit0,M5) )
                 => ( ( Xa2 = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa2 = aa(num,num,bit0,N5) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa(num,num,bit1,N5) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N5)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit1,M5) )
                       => ( ( Xa2 = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa2 = aa(num,num,bit0,N5) )
                             => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_pp(num,option(num)),bit_and_not_num(M5,N5)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa(num,num,bit1,N5) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_3975_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,X,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_3976_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),I2: B,B4: set(A),J5: 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),A5,I2,B4)),J5)) = 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),A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),bot_bot(set(B))))))),if(set(A),aa(set(B),bool,member(B,I2),J5),B4,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_3977_image__update,axiom,
    ! [B: $tType,A: $tType,X: A,A5: set(A),F3: fun(A,B),N2: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),A5))
     => ( aa(set(A),set(B),image2(A,B,fun_upd(A,B,F3,X,N2)),A5) = aa(set(A),set(B),image2(A,B,F3),A5) ) ) ).

% image_update
tff(fact_3978_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xo: option(B),Y: A] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),Xo) = aa(A,option(A),some(A),Y) )
    <=> ? [Z6: B] :
          ( ( Xo = aa(B,option(B),some(B),Z6) )
          & ( aa(B,A,F3,Z6) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_3979_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: option(B)] :
      ( ( none(A) = aa(option(B),option(A),map_option(B,A,F3),X) )
    <=> ( X = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_3980_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_3981_option_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: option(A)] :
      ( ( aa(option(A),option(B),map_option(A,B,F3),A3) = none(B) )
    <=> ( A3 = none(A) ) ) ).

% option.map_disc_iff
tff(fact_3982_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,B),X5: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),aTP_Lamp_qt(A,option(C))),X5) = none(B) ).

% map_option_o_empty
tff(fact_3983_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: A,H: fun(B,A),F3: fun(C,B),X: option(C)] : case_option(A,B,G3,H,aa(option(C),option(B),map_option(C,B,F3),X)) = case_option(A,C,G3,aa(fun(C,B),fun(C,A),comp(B,A,C,H),F3),X) ).

% case_map_option
tff(fact_3984_option_Omap__ident,axiom,
    ! [A: $tType,T6: option(A)] : aa(option(A),option(A),map_option(A,A,aTP_Lamp_cq(A,A)),T6) = T6 ).

% option.map_ident
tff(fact_3985_option_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X2: A] : aa(option(A),option(B),map_option(A,B,F3),aa(A,option(A),some(A),X2)) = aa(B,option(B),some(B),aa(A,B,F3,X2)) ).

% option.simps(9)
tff(fact_3986_map__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y: option(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( X = Y )
     => ( ! [A6: A] :
            ( ( Y = aa(A,option(A),some(A),A6) )
           => ( aa(A,B,F3,A6) = aa(A,B,G3,A6) ) )
       => ( aa(option(A),option(B),map_option(A,B,F3),X) = aa(option(A),option(B),map_option(A,B,G3),Y) ) ) ) ).

% map_option_cong
tff(fact_3987_option_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B)] : aa(option(A),option(B),map_option(A,B,F3),none(A)) = none(B) ).

% option.simps(8)
tff(fact_3988_map__option_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(B,C),G3: fun(A,B),Option: option(A)] : aa(option(B),option(C),map_option(B,C,F3),aa(option(A),option(B),map_option(A,B,G3),Option)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G3)),Option) ).

% map_option.compositionality
tff(fact_3989_option_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G3: fun(B,C),F3: fun(A,B),V2: option(A)] : aa(option(B),option(C),map_option(B,C,G3),aa(option(A),option(B),map_option(A,B,F3),V2)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G3),F3)),V2) ).

% option.map_comp
tff(fact_3990_map__option_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C),G3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),comp(option(B),option(C),option(A),map_option(B,C,F3)),map_option(A,B,G3)) = map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G3)) ).

% map_option.comp
tff(fact_3991_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,nat),G3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),comp(option(B),nat,option(A),size_option(B,F3)),map_option(A,B,G3)) = size_option(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F3),G3)) ).

% option.size_gen_o_map
tff(fact_3992_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C2: B,P: fun(fun(A,B),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_qu(fun(A,B),fun(B,fun(A,bool)),F3),C2))))
     => ( pp(aa(fun(A,B),bool,P,aTP_Lamp_qf(B,fun(A,B),C2)))
       => ( ! [A6: A,B5: B,F2: fun(A,B)] :
              ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_qv(B,fun(fun(A,B),fun(A,bool)),C2),F2))))
             => ( ( aa(A,B,F2,A6) = C2 )
               => ( ( B5 != C2 )
                 => ( pp(aa(fun(A,B),bool,P,F2))
                   => pp(aa(fun(A,B),bool,P,fun_upd(A,B,F2,A6,B5))) ) ) ) )
         => pp(aa(fun(A,B),bool,P,F3)) ) ) ) ).

% finite_update_induct
tff(fact_3993_set__remove1__subset,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,X,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_remove1_subset
tff(fact_3994_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A3,Xs)) ) ) ).

% sorted_remove1
tff(fact_3995_ran__map__option,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),M: fun(B,option(C))] : ran(B,A,aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_qw(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),F3),M)) = aa(set(C),set(A),image2(C,A,F3),ran(B,C,M)) ).

% ran_map_option
tff(fact_3996_map__option__case,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Y: option(B)] : aa(option(B),option(A),map_option(B,A,F3),Y) = case_option(option(A),B,none(A),aTP_Lamp_qx(fun(B,A),fun(B,option(A)),F3),Y) ).

% map_option_case
tff(fact_3997_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,X: B,A5: set(B),F3: fun(B,A),Y: A] :
      ( ( pp(aa(set(B),bool,member(B,X),A5))
       => ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F3,X,Y)),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) )
      & ( ~ pp(aa(set(B),bool,member(B,X),A5))
       => ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F3,X,Y)),A5) = aa(set(B),set(A),image2(B,A,F3),A5) ) ) ) ).

% fun_upd_image
tff(fact_3998_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = remove1(A,X,aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_3999_and__not__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2)))
       => ( ( ( X = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = none(num) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [N5: num] :
                  ( ( Xa2 = aa(num,num,bit0,N5) )
                 => ( ( Y = aa(num,option(num),some(num),one2) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N5)))) ) ) )
           => ( ( ( X = one2 )
               => ! [N5: num] :
                    ( ( Xa2 = aa(num,num,bit1,N5) )
                   => ( ( Y = none(num) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N5)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa(num,num,bit0,N5) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N5)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N5)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa2 = aa(num,num,bit1,N5) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N5)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa(num,num,bit0,N5) )
                               => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_pp(num,option(num)),bit_and_not_num(M5,N5)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N5)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N5) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N5)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N5)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_4000_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != aa(num,option(num),some(num),one2) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N5: num] : Xa2 = aa(num,num,bit0,N5)
             => ( Y != none(num) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N5: num] : Xa2 = aa(num,num,bit1,N5)
               => ( Y != aa(num,option(num),some(num),one2) ) ) )
           => ( ( ? [M5: num] : X = aa(num,num,bit0,M5)
               => ( ( Xa2 = one2 )
                 => ( Y != none(num) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa2 = aa(num,num,bit0,N5) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa(num,num,bit1,N5) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N5)) ) ) )
                 => ( ( ? [M5: num] : X = aa(num,num,bit1,M5)
                     => ( ( Xa2 = one2 )
                       => ( Y != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa2 = aa(num,num,bit0,N5) )
                             => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N5)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa(num,num,bit1,N5) )
                               => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_pp(num,option(num)),bit_un7362597486090784418nd_num(M5,N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_4001_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un2480387367778600638or_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ! [N5: num] :
                ( ( Xa2 = aa(num,num,bit0,N5) )
               => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,N5)) ) ) )
         => ( ( ( X = one2 )
             => ! [N5: num] :
                  ( ( Xa2 = aa(num,num,bit1,N5) )
                 => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,N5)) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa(num,num,bit0,M5) )
                 => ( ( Xa2 = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa2 = aa(num,num,bit0,N5) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa(num,num,bit1,N5) )
                         => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N5))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit1,M5) )
                       => ( ( Xa2 = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa2 = aa(num,num,bit0,N5) )
                             => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N5))) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa(num,num,bit1,N5) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_4002_execute__lookup,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),H: 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,R3)),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)),ref_get(A,H,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),one_one(nat)))) ) ).

% execute_lookup
tff(fact_4003_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,X: A,X5: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_bk(A,option(B)),X,none(B)),X5) = none(B) ).

% empty_upd_none
tff(fact_4004_image__map__upd,axiom,
    ! [B: $tType,A: $tType,X: A,A5: set(A),M: fun(A,option(B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),A5))
     => ( aa(set(A),set(option(B)),image2(A,option(B),fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y))),A5) = aa(set(A),set(option(B)),image2(A,option(B),M),A5) ) ) ).

% image_map_upd
tff(fact_4005_map__option__o__map__upd,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),M: fun(A,option(C)),A3: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),fun_upd(A,option(C),M,A3,aa(C,option(C),some(C),B2))) = fun_upd(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),M),A3,aa(B,option(B),some(B),aa(C,B,F3,B2))) ).

% map_option_o_map_upd
tff(fact_4006_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M,A3) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M,A3,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),ran(B,A,M)) ) ) ).

% ran_map_upd
tff(fact_4007_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,X: B,N2: fun(A,option(B)),Y: B] :
      ( ( fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),X)) = fun_upd(A,option(B),N2,A3,aa(B,option(B),some(B),Y)) )
     => ( X = Y ) ) ).

% map_upd_eqD1
tff(fact_4008_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,T6: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),T6,K) = aa(A,option(A),some(A),X) )
     => ( fun_upd(B,option(A),T6,K,aa(A,option(A),some(A),X)) = T6 ) ) ).

% map_upd_triv
tff(fact_4009_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),A3: B,B2: A,X: B,Y: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),M,A3,aa(A,option(A),some(A),B2)),X) = aa(A,option(A),some(A),Y) )
    <=> ( ( ( X = A3 )
          & ( B2 = Y ) )
        | ( ( X != A3 )
          & ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_4010_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,T6: fun(A,option(B)),K: A,X: B] :
      ~ ! [X3: A] : aa(A,option(B),fun_upd(A,option(B),T6,K,aa(B,option(B),some(B),X)),X3) = none(B) ).

% map_upd_nonempty
tff(fact_4011_and__num_Osimps_I1_J,axiom,
    bit_un7362597486090784418nd_num(one2,one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(1)
tff(fact_4012_and__num_Osimps_I3_J,axiom,
    ! [N2: num] : bit_un7362597486090784418nd_num(one2,aa(num,num,bit1,N2)) = aa(num,option(num),some(num),one2) ).

% and_num.simps(3)
tff(fact_4013_and__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(7)
tff(fact_4014_and__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N2: num,Q3: num] :
          ( ( bit_un7362597486090784418nd_num(M,N2) = aa(num,option(num),some(num),Q3) )
        <=> ( bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% and_num_eq_Some_iff
tff(fact_4015_xor__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N2: num,Q3: num] :
          ( ( bit_un2480387367778600638or_num(M,N2) = aa(num,option(num),some(num),Q3) )
        <=> ( bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% xor_num_eq_Some_iff
tff(fact_4016_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),A3: B,B2: A] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F3),top_top(set(B)))))
     => pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),fun_upd(B,option(A),F3,A3,aa(A,option(A),some(A),B2))),top_top(set(B))))) ) ).

% finite_range_updI
tff(fact_4017_lookup__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A)] : ref_lookup(A,R3) = heap_Time_tap(A,aTP_Lamp_qy(ref(A),fun(heap_ext(product_unit),A),R3)) ) ).

% lookup_def
tff(fact_4018_xor__num_Osimps_I2_J,axiom,
    ! [N2: num] : bit_un2480387367778600638or_num(one2,aa(num,num,bit0,N2)) = aa(num,option(num),some(num),aa(num,num,bit1,N2)) ).

% xor_num.simps(2)
tff(fact_4019_xor__num_Osimps_I3_J,axiom,
    ! [N2: num] : bit_un2480387367778600638or_num(one2,aa(num,num,bit1,N2)) = aa(num,option(num),some(num),aa(num,num,bit0,N2)) ).

% xor_num.simps(3)
tff(fact_4020_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_un2480387367778600638or_num(aa(num,num,bit0,M),one2) = aa(num,option(num),some(num),aa(num,num,bit1,M)) ).

% xor_num.simps(4)
tff(fact_4021_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_un2480387367778600638or_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% xor_num.simps(7)
tff(fact_4022_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N2: num] : bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(M,N2)) ) ).

% numeral_and_num
tff(fact_4023_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N2: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(M,N2)) ) ).

% numeral_xor_num
tff(fact_4024_and__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),aa(num,num,bit1,N2)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_pp(num,option(num)),bit_un7362597486090784418nd_num(M,N2)) ).

% and_num.simps(9)
tff(fact_4025_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] : bit_un2480387367778600638or_num(aa(num,num,bit0,M),aa(num,num,bit1,N2)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M,N2))) ).

% xor_num.simps(6)
tff(fact_4026_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] : bit_un2480387367778600638or_num(aa(num,num,bit1,M),aa(num,num,bit0,N2)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M,N2))) ).

% xor_num.simps(8)
tff(fact_4027_and__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2)))
       => ( ( ( X = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = aa(num,option(num),some(num),one2) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [N5: num] :
                  ( ( Xa2 = aa(num,num,bit0,N5) )
                 => ( ( Y = none(num) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N5)))) ) ) )
           => ( ( ( X = one2 )
               => ! [N5: num] :
                    ( ( Xa2 = aa(num,num,bit1,N5) )
                   => ( ( Y = aa(num,option(num),some(num),one2) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N5)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = none(num) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa(num,num,bit0,N5) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N5)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N5)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa2 = aa(num,num,bit1,N5) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N5)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),one2) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa(num,num,bit0,N5) )
                               => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N5)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N5)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N5) )
                                 => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_pp(num,option(num)),bit_un7362597486090784418nd_num(M5,N5)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N5)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_4028_xor__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un2480387367778600638or_num(X,Xa2) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2)))
       => ( ( ( X = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = none(num) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [N5: num] :
                  ( ( Xa2 = aa(num,num,bit0,N5) )
                 => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,N5)) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N5)))) ) ) )
           => ( ( ( X = one2 )
               => ! [N5: num] :
                    ( ( Xa2 = aa(num,num,bit1,N5) )
                   => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,N5)) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N5)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,M5)) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa2 = aa(num,num,bit0,N5) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N5)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N5)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa2 = aa(num,num,bit1,N5) )
                           => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N5))) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N5)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa2 = aa(num,num,bit0,N5) )
                               => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N5))) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N5)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N5) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N5)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N5)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_4029_sngr__assn__raw_Oelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb))
         => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
                ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
               => ( ( ref_get(A,H4,X) = Xa2 )
                  & ( As4 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H4))) ) ) ) ) ).

% sngr_assn_raw.elims(3)
tff(fact_4030_sngr__assn__raw_Oelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb))
         => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
                ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
               => ~ ( ( ref_get(A,H4,X) = Xa2 )
                    & ( As4 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H4))) ) ) ) ) ).

% sngr_assn_raw.elims(2)
tff(fact_4031_relH__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [As: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: ref(A)] :
          ( relH(As,H,H3)
         => ( pp(aa(set(nat),bool,member(nat,addr_of_ref(A,R3)),As))
           => ( ref_get(A,H,R3) = ref_get(A,H3,R3) ) ) ) ) ).

% relH_ref
tff(fact_4032_sngr__assn__raw_Osimps,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),X: A,H: heap_ext(product_unit),As: set(nat)] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,R3,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)))
        <=> ( ( ref_get(A,H,R3) = X )
            & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,R3)),bot_bot(set(nat))) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,R3)),lim(product_unit,H))) ) ) ) ).

% sngr_assn_raw.simps
tff(fact_4033_sngr__assn__raw_Oelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
          ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb))
          <=> pp(Y) )
         => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
                ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
               => ( pp(Y)
                <=> ~ ( ( ref_get(A,H4,X) = Xa2 )
                      & ( As4 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H4))) ) ) ) ) ) ).

% sngr_assn_raw.elims(1)
tff(fact_4034_sngr__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
          ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb))
          <=> pp(Y) )
         => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,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)))),X),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))),Xa2),Xb))))
           => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
                  ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
                 => ( ( pp(Y)
                    <=> ( ( ref_get(A,H4,X) = Xa2 )
                        & ( As4 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H4))) ) )
                   => ~ pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,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)))),X),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))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))))) ) ) ) ) ) ).

% sngr_assn_raw.pelims(1)
tff(fact_4035_sngr__assn__raw_Opelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb))
         => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,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)))),X),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))),Xa2),Xb))))
           => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
                  ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
                 => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,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)))),X),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))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))))
                   => ~ ( ( ref_get(A,H4,X) = Xa2 )
                        & ( As4 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H4))) ) ) ) ) ) ) ).

% sngr_assn_raw.pelims(2)
tff(fact_4036_sngr__assn__raw_Opelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb))
         => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,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)))),X),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))),Xa2),Xb))))
           => ~ ! [H4: heap_ext(product_unit),As4: set(nat)] :
                  ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4) )
                 => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,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)))),X),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))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4)))))
                   => ( ( ref_get(A,H4,X) = Xa2 )
                      & ( As4 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H4))) ) ) ) ) ) ) ).

% sngr_assn_raw.pelims(3)
tff(fact_4037_option_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: C,Ga: fun(B,C),F3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),comp(option(B),C,option(A),rec_option(C,B,G3,Ga)),map_option(A,B,F3)) = rec_option(C,A,G3,aa(fun(A,B),fun(A,C),aTP_Lamp_be(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F3)) ).

% option.rec_o_map
tff(fact_4038_option_Osimps_I7_J,axiom,
    ! [C: $tType,A: $tType,F1: C,F22: fun(A,C),X2: A] : aa(option(A),C,rec_option(C,A,F1,F22),aa(A,option(A),some(A),X2)) = aa(A,C,F22,X2) ).

% option.simps(7)
tff(fact_4039_option_Osimps_I6_J,axiom,
    ! [A: $tType,C: $tType,F1: C,F22: fun(A,C)] : aa(option(A),C,rec_option(C,A,F1,F22),none(A)) = F1 ).

% option.simps(6)
tff(fact_4040_subset__mset_Osum__list__update,axiom,
    ! [A: $tType,K: nat,Xs: list(multiset(A)),X: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Xs)))
     => ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),list_update(multiset(A),Xs,K,X)) = 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)),Xs)),X)),aa(nat,multiset(A),nth(multiset(A),Xs),K)) ) ) ).

% subset_mset.sum_list_update
tff(fact_4041_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ra(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% times_int.abs_eq
tff(fact_4042_set__removeAll,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_4043_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ~ ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),S))
                  & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X5)),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S)))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4044_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X3: nat,Y3: nat] : Z2 != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),Y3)) ).

% eq_Abs_Integ
tff(fact_4045_length__removeAll__less__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_removeAll_less_eq
tff(fact_4046_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_4047_int__def,axiom,
    ! [N2: nat] : aa(nat,int,semiring_1_of_nat(int),N2) = 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),N2),zero_zero(nat))) ).

% int_def
tff(fact_4048_length__removeAll__less,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_removeAll_less
tff(fact_4049_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_rb(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_4050_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => pp(aa(set(A),bool,member(A,lattic7623131987881927897min_on(A,B,F3,S)),S)) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_4051_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_4052_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: product_prod(nat,nat)] : aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rc(nat,fun(nat,A))),X) ) ).

% of_int.abs_eq
tff(fact_4053_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_re(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_4054_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rg(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_4055_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ri(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_4056_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_4057_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Y: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,member(A,Y),S))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S))),aa(A,B,F3,Y))) ) ) ) ) ).

% arg_min_least
tff(fact_4058_subset__mset_Oelem__le__sum__list,axiom,
    ! [A: $tType,K: nat,Ns: list(multiset(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Ns)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),nth(multiset(A),Ns),K)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns))) ) ).

% subset_mset.elem_le_sum_list
tff(fact_4059_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F3: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_rl(fun(B,list(A)),fun(B,set(A)),F3)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_4060_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4061_execute__change,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [F3: fun(A,A),R3: ref(A),H: 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,F3,R3)),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)),aa(A,A,F3,ref_get(A,H,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),ref_set(A,R3,aa(A,A,F3,ref_get(A,H,R3)),H)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))) ) ).

% execute_change
tff(fact_4062_lim__set,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),V2: A,H: heap_ext(product_unit)] : lim(product_unit,ref_set(A,R3,V2,H)) = lim(product_unit,H) ) ).

% lim_set
tff(fact_4063_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4064_mset__le__subtract,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),B4))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A5),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B4),C4))) ) ).

% mset_le_subtract
tff(fact_4065_subset__mset_Ofinite__has__minimal,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ? [X3: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),A5))
            & ! [Xa3: multiset(A)] :
                ( pp(aa(set(multiset(A)),bool,member(multiset(A),Xa3),A5))
               => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xa3),X3))
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% subset_mset.finite_has_minimal
tff(fact_4066_subset__mset_Ofinite__has__maximal,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ? [X3: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),A5))
            & ! [Xa3: multiset(A)] :
                ( pp(aa(set(multiset(A)),bool,member(multiset(A),Xa3),A5))
               => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X3),Xa3))
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% subset_mset.finite_has_maximal
tff(fact_4067_mset__le__addE,axiom,
    ! [A: $tType,Xs: multiset(A),Ys2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xs),Ys2))
     => ~ ! [Zs2: multiset(A)] : Ys2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xs),Zs2) ) ).

% mset_le_addE
tff(fact_4068_mset__le__distrib,axiom,
    ! [A: $tType,X6: multiset(A),A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4)))
     => ~ ! [Xa5: multiset(A),Xb4: multiset(A)] :
            ( ( X6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa5),Xb4) )
           => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xa5),A5))
             => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xb4),B4)) ) ) ) ).

% mset_le_distrib
tff(fact_4069_mset__union__subset,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4)),C4))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),C4))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B4),C4)) ) ) ).

% mset_union_subset
tff(fact_4070_mset__le__decr__left1,axiom,
    ! [A: $tType,A3: multiset(A),C2: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% mset_le_decr_left1
tff(fact_4071_mset__le__decr__left2,axiom,
    ! [A: $tType,C2: multiset(A),A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% mset_le_decr_left2
tff(fact_4072_mset__le__incr__right1,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2))) ) ).

% mset_le_incr_right1
tff(fact_4073_mset__le__incr__right2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2))) ) ).

% mset_le_incr_right2
tff(fact_4074_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),Xs)) ) ).

% insort_left_comm
tff(fact_4075_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4076_subset__mset_Olift__Suc__mono__le,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N2: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,N5)),aa(nat,multiset(A),F3,aa(nat,nat,suc,N5))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N6))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,N2)),aa(nat,multiset(A),F3,N6))) ) ) ).

% subset_mset.lift_Suc_mono_le
tff(fact_4077_subset__mset_Olift__Suc__antimono__le,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N2: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,aa(nat,nat,suc,N5))),aa(nat,multiset(A),F3,N5)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N6))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,N6)),aa(nat,multiset(A),F3,N2))) ) ) ).

% subset_mset.lift_Suc_antimono_le
tff(fact_4078_mset__le__subtract__left,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),A5)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),X6)) ) ) ).

% mset_le_subtract_left
tff(fact_4079_mset__le__subtract__right,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),B4)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B4),X6)) ) ) ).

% mset_le_subtract_right
tff(fact_4080_size__mset__mono,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),B4))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),A5)),aa(multiset(A),nat,size_size(multiset(A)),B4))) ) ).

% size_mset_mono
tff(fact_4081_subset__mset_OcInf__eq__non__empty,axiom,
    ! [A: $tType,X6: set(multiset(A)),A3: multiset(A)] :
      ( ( X6 != bot_bot(set(multiset(A))) )
     => ( ! [X3: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),X6))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),X3)) )
       => ( ! [Y3: multiset(A)] :
              ( ! [X5: multiset(A)] :
                  ( pp(aa(set(multiset(A)),bool,member(multiset(A),X5),X6))
                 => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Y3),X5)) )
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Y3),A3)) )
         => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X6) = A3 ) ) ) ) ).

% subset_mset.cInf_eq_non_empty
tff(fact_4082_subset__mset_OcInf__greatest,axiom,
    ! [A: $tType,X6: set(multiset(A)),Z2: multiset(A)] :
      ( ( X6 != bot_bot(set(multiset(A))) )
     => ( ! [X3: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),X6))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Z2),X3)) )
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Z2),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X6))) ) ) ).

% subset_mset.cInf_greatest
tff(fact_4083_subset__mset_OcSup__least,axiom,
    ! [A: $tType,X6: set(multiset(A)),Z2: multiset(A)] :
      ( ( X6 != bot_bot(set(multiset(A))) )
     => ( ! [X3: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),X6))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X3),Z2)) )
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X6)),Z2)) ) ) ).

% subset_mset.cSup_least
tff(fact_4084_subset__mset_OcSup__eq__non__empty,axiom,
    ! [A: $tType,X6: set(multiset(A)),A3: multiset(A)] :
      ( ( X6 != bot_bot(set(multiset(A))) )
     => ( ! [X3: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),X6))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X3),A3)) )
       => ( ! [Y3: multiset(A)] :
              ( ! [X5: multiset(A)] :
                  ( pp(aa(set(multiset(A)),bool,member(multiset(A),X5),X6))
                 => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X5),Y3)) )
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),Y3)) )
         => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X6) = A3 ) ) ) ) ).

% subset_mset.cSup_eq_non_empty
tff(fact_4085_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_4086_subset__mset_OcINF__greatest,axiom,
    ! [A: $tType,B: $tType,A5: set(B),M: multiset(A),F3: fun(B,multiset(A))] :
      ( ( A5 != bot_bot(set(B)) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A5))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),aa(B,multiset(A),F3,X3))) )
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5)))) ) ) ).

% subset_mset.cINF_greatest
tff(fact_4087_subset__mset_OcSUP__least,axiom,
    ! [B: $tType,A: $tType,A5: set(B),F3: fun(B,multiset(A)),M6: multiset(A)] :
      ( ( A5 != bot_bot(set(B)) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A5))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,X3)),M6)) )
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),M6)) ) ) ).

% subset_mset.cSUP_least
tff(fact_4088_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,member(A,A3),aa(list(A),set(A),set2(A),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),A3),remove1(A,A3,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_4089_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4090_relH__set__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),As: set(nat),H: heap_ext(product_unit),X: A] :
          ( ~ pp(aa(set(nat),bool,member(nat,addr_of_ref(A,R3)),As))
         => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
           => relH(As,H,ref_set(A,R3,X,H)) ) ) ) ).

% relH_set_ref
tff(fact_4091_change__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [F3: fun(A,A),R3: ref(A)] : ref_change(A,F3,R3) = heap_Time_bind(A,A,ref_lookup(A,R3),aa(ref(A),fun(A,heap_Time_Heap(A)),aTP_Lamp_ro(fun(A,A),fun(ref(A),fun(A,heap_Time_Heap(A))),F3),R3)) ) ).

% change_def
tff(fact_4092_subset__mset_Osum__mono2,axiom,
    ! [A: $tType,B: $tType,B4: set(B),A5: set(B),F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),B4))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
       => ( ! [B5: B] :
              ( pp(aa(set(B),bool,member(B,B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A5)))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,B5))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,A5)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,B4))) ) ) ) ).

% subset_mset.sum_mono2
tff(fact_4093_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rg(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_4094_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),Xa2))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_re(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_4095_subset__mset_Osum__mono,axiom,
    ! [A: $tType,B: $tType,K5: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( ! [I3: B] :
          ( pp(aa(set(B),bool,member(B,I3),K5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,I3)),aa(B,multiset(A),G3,I3))) )
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,K5)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),G3,K5))) ) ).

% subset_mset.sum_mono
tff(fact_4096_subset__mset_Osum__nonneg__0,axiom,
    ! [B: $tType,A: $tType,S2: set(B),F3: fun(B,multiset(A)),I2: B] :
      ( pp(aa(set(B),bool,finite_finite2(B),S2))
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),S2))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I3))) )
       => ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,S2) = zero_zero(multiset(A)) )
         => ( pp(aa(set(B),bool,member(B,I2),S2))
           => ( aa(B,multiset(A),F3,I2) = zero_zero(multiset(A)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_0
tff(fact_4097_subset__mset_Osum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType,S2: set(B),F3: fun(B,multiset(A)),B4: multiset(A),I2: B] :
      ( pp(aa(set(B),bool,finite_finite2(B),S2))
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),S2))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I3))) )
       => ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,S2) = B4 )
         => ( pp(aa(set(B),bool,member(B,I2),S2))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,I2)),B4)) ) ) ) ) ).

% subset_mset.sum_nonneg_leq_bound
tff(fact_4098_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),X) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rc(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).

% of_int.rep_eq
tff(fact_4099_mset__size2elem,axiom,
    ! [A: $tType,P: multiset(A),Q3: A,Q7: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q7),zero_zero(multiset(A))))),P))
       => ( P = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q7),zero_zero(multiset(A)))) ) ) ) ).

% mset_size2elem
tff(fact_4100_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_rq(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_4101_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_rb(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4102_pred__nat__def,axiom,
    pred_nat = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_rr(nat,fun(nat,bool)))) ).

% pred_nat_def
tff(fact_4103_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R3: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),lex_prod(A,B,R3,S2)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A4)),R3))
        | ( ( A3 = A4 )
          & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B3)),S2)) ) ) ) ).

% in_lex_prod
tff(fact_4104_mset__union__2__elem,axiom,
    ! [A: $tType,A3: A,B2: A,C2: A,M6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),M6) )
     => ( ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = M6 )
          & ( B2 = C2 ) )
        | ( ( A3 = C2 )
          & ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))) = M6 ) ) ) ) ).

% mset_union_2_elem
tff(fact_4105_mset__le__add__mset__decr__left1,axiom,
    ! [A: $tType,C2: A,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),A3)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% mset_le_add_mset_decr_left1
tff(fact_4106_mset__le__add__mset,axiom,
    ! [A: $tType,X: A,B4: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),B4)),C4))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))),C4))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B4),C4)) ) ) ).

% mset_le_add_mset
tff(fact_4107_mset__le__single__cases,axiom,
    ! [A: $tType,M6: multiset(A),A3: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))))
     => ( ( M6 != zero_zero(multiset(A)) )
       => ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ) ).

% mset_le_single_cases
tff(fact_4108_mset__le__add__mset__decr__left2,axiom,
    ! [A: $tType,C2: A,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),A3)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),zero_zero(multiset(A)))),B2)) ) ).

% mset_le_add_mset_decr_left2
tff(fact_4109_mset__le__subtract__add__mset__left,axiom,
    ! [A: $tType,X: A,B4: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),B4)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A))))))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))),X6)) ) ) ).

% mset_le_subtract_add_mset_left
tff(fact_4110_mset__le__subtract__add__mset__right,axiom,
    ! [A: $tType,X: A,B4: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),B4)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),B4)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B4),X6)) ) ) ).

% mset_le_subtract_add_mset_right
tff(fact_4111_mset__single__cases,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R6: A,C9: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C9) )
     => ( ( ( S2 = R6 )
         => ( C2 != C9 ) )
       => ~ ( ( C9 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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)),C9),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A))))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C9),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases
tff(fact_4112_mset__single__cases_H,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R6: A,C9: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C9) )
     => ( ( ( S2 = R6 )
         => ( C2 != C9 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C9 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))),Cc) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C9),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),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),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases'
tff(fact_4113_mset__single__cases2,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R6: A,C9: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C9) )
     => ( ( ( S2 = R6 )
         => ( C2 != C9 ) )
       => ~ ( ( C9 = 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)),C9),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C9),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases2
tff(fact_4114_mset__single__cases2_H,axiom,
    ! [A: $tType,S2: A,C2: multiset(A),R6: A,C9: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C9) )
     => ( ( ( S2 = R6 )
         => ( C2 != C9 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C9 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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)),Cc),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C9),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),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),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases2'
tff(fact_4115_mset__unplusm__dist__cases,axiom,
    ! [A: $tType,S2: A,A5: multiset(A),B4: multiset(A),C4: 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(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),A5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),C4) )
     => ( ( ( B4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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)),B4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
         => ( A5 != 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)),B4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))),C4) ) )
       => ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
           => ( A5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases
tff(fact_4116_mset__unplusm__dist__cases2,axiom,
    ! [A: $tType,B4: multiset(A),C4: multiset(A),S2: A,A5: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),A5) )
     => ( ( ( B4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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)),B4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
         => ( A5 != 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)),B4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))),C4) ) )
       => ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),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)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
           => ( A5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases2
tff(fact_4117_size__Diff1__le,axiom,
    ! [A: $tType,M6: multiset(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(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)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ).

% size_Diff1_le
tff(fact_4118_mset__size__le1__cases,axiom,
    ! [A: $tType,M6: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),M6)),aa(nat,nat,suc,zero_zero(nat))))
     => ( ( M6 != zero_zero(multiset(A)) )
       => ~ ! [M5: A] : M6 != aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M5),zero_zero(multiset(A))) ) ) ).

% mset_size_le1_cases
tff(fact_4119_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_ra(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4120_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_rk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4121_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_ri(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4122_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),R2: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R2) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_rt(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),P),R2)))) ).

% same_fst_def
tff(fact_4123_same__fstI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),X: A,Y7: B,Y: B,R2: fun(A,set(product_prod(B,B)))] :
      ( pp(aa(A,bool,P,X))
     => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y)),aa(A,set(product_prod(B,B)),R2,X)))
       => pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y))),same_fst(A,B,P,R2))) ) ) ).

% same_fstI
tff(fact_4124_sorted__list__of__multiset__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,M6: multiset(A)] : linord6283353356039996273ltiset(A,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),M6)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),linord6283353356039996273ltiset(A,M6)) ) ).

% sorted_list_of_multiset_insert
tff(fact_4125_merge__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L1: list(A),L22: list(A)] :
          ( ( distinct(A,L1)
            & sorted_wrt(A,ord_less_eq(A),L1) )
         => ( ( distinct(A,L22)
              & sorted_wrt(A,ord_less_eq(A),L22) )
           => ( distinct(A,merge(A,L1,L22))
              & sorted_wrt(A,ord_less_eq(A),merge(A,L1,L22))
              & ( aa(list(A),set(A),set2(A),merge(A,L1,L22)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),L1)),aa(list(A),set(A),set2(A),L22)) ) ) ) ) ) ).

% merge_correct
tff(fact_4126_distinct__foldl__invar,axiom,
    ! [B: $tType,A: $tType,S: list(A),I5: fun(set(A),fun(B,bool)),Sigma_0: B,F3: fun(B,fun(A,B))] :
      ( distinct(A,S)
     => ( pp(aa(B,bool,aa(set(A),fun(B,bool),I5,aa(list(A),set(A),set2(A),S)),Sigma_0))
       => ( ! [X3: A,It: set(A),Sigma: B] :
              ( pp(aa(set(A),bool,member(A,X3),It))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S)))
               => ( pp(aa(B,bool,aa(set(A),fun(B,bool),I5,It),Sigma))
                 => pp(aa(B,bool,aa(set(A),fun(B,bool),I5,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)),insert(A),X3),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F3,Sigma),X3))) ) ) )
         => pp(aa(B,bool,aa(set(A),fun(B,bool),I5,bot_bot(set(A))),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),Sigma_0),S))) ) ) ) ).

% distinct_foldl_invar
tff(fact_4127_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
         => ( ( pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1962203154675924110t_prod(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = groups1962203154675924110t_prod(B,A,P3,I5) ) )
            & ( ~ pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1962203154675924110t_prod(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P3,I2)),groups1962203154675924110t_prod(B,A,P3,I5)) ) ) ) ) ) ).

% prod.insert'
tff(fact_4128_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A)] : groups1962203154675924110t_prod(B,A,P3,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_4129_foldl__length,axiom,
    ! [A: $tType,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_ru(nat,fun(A,nat))),zero_zero(nat)),L) = aa(list(A),nat,size_size(list(A)),L) ).

% foldl_length
tff(fact_4130_foldl__A1__eq,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),N2: A,I2: A,Ww: list(A)] :
      ( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F3,N2),E2) = E2
     => ( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F3,E2),N2) = E2
       => ( ! [A6: A,B5: A,C3: A] : aa(A,A,aa(A,fun(A,A),F3,A6),aa(A,A,aa(A,fun(A,A),F3,B5),C3)) = aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,A6),B5)),C3)
         => ( aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F3),I2),Ww) = aa(A,A,aa(A,fun(A,A),F3,I2),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F3),N2),Ww)) ) ) ) ) ).

% foldl_A1_eq
tff(fact_4131_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),I5: set(B)] : groups1962203154675924110t_prod(B,A,G3,aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_rv(fun(B,A),fun(set(B),fun(B,bool)),G3),I5))) = groups1962203154675924110t_prod(B,A,G3,I5) ) ).

% prod.non_neutral'
tff(fact_4132_foldl__absorb1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Zs: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,aa(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),foldl(A,A,times_times(A)),X),Zs) ) ).

% foldl_absorb1
tff(fact_4133_foldl__un__empty__eq,axiom,
    ! [A: $tType,I2: set(A),Ww: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),I2),Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I2),aa(list(set(A)),set(A),aa(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_4134_prod_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_en(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G3,I5)),groups1962203154675924110t_prod(B,A,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_4135_prod_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T2: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,G3,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G3,S) = groups1962203154675924110t_prod(B,A,G3,T2) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_4136_prod_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T2: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,G3,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G3,T2) = groups1962203154675924110t_prod(B,A,G3,S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_4137_prod_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T2: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,H,I3) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G3,S) = groups1962203154675924110t_prod(B,A,H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_4138_prod_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T2: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T2))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),S)))
               => ( aa(B,A,G3,X3) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G3,T2) = groups1962203154675924110t_prod(B,A,H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_4139_foldl__set,axiom,
    ! [A: $tType,L: list(set(A))] : aa(list(set(A)),set(A),aa(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),bool),set(set(A)),collect(set(A)),aTP_Lamp_rw(list(set(A)),fun(set(A),bool),L))) ).

% foldl_set
tff(fact_4140_foldl__length__aux,axiom,
    ! [A: $tType,A3: nat,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_ru(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_4141_prod_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),G3))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_en(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G3,I5)),groups1962203154675924110t_prod(B,A,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_4142_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P3: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1962203154675924110t_prod(B,A,P3,I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1962203154675924110t_prod(B,A,P3,I5) = one_one(A) ) ) ) ) ).

% prod.G_def
tff(fact_4143_set__n__lists,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N2,Xs)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),bool),aTP_Lamp_rx(nat,fun(list(A),fun(list(A),bool)),N2),Xs)) ).

% set_n_lists
tff(fact_4144_Gcd__remove0__nat,axiom,
    ! [M6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M6))
     => ( gcd_Gcd(nat,M6) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M6),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_4145_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_rm(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_4146_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => ( num_of_nat(aa(nat,nat,suc,N2)) = inc(num_of_nat(N2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => ( num_of_nat(aa(nat,nat,suc,N2)) = one2 ) ) ) ).

% num_of_nat.simps(2)
tff(fact_4147_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_4148_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( gcd_Gcd(A,A5) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_4149_Gcd__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Gcd(A,aa(set(B),set(A),image2(B,A,F3),A5))),gcd_Gcd(A,aa(set(B),set(A),image2(B,A,G3),A5)))) ) ) ).

% Gcd_mono
tff(fact_4150_subset__subseqs,axiom,
    ! [A: $tType,X6: set(A),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(set(set(A)),bool,member(set(A),X6),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))))) ) ).

% subset_subseqs
tff(fact_4151_Un__set__drop__extend,axiom,
    ! [A: $tType,J: nat,L: list(set(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(set(A)),nat,size_size(list(set(A))),L)))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(nat,set(A),nth(set(A),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),J,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).

% Un_set_drop_extend
tff(fact_4152_Gcd__eq__Max,axiom,
    ! [M6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M6))
     => ( ( M6 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
         => ( gcd_Gcd(nat,M6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),aTP_Lamp_ry(nat,set(nat))),M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_4153_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_rz(nat,bool))) ) ).

% semiring_char_def
tff(fact_4154_Gcd__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,aa(set(int),set(int),image2(int,int,abs_abs(int)),K5)) = gcd_Gcd(int,K5) ).

% Gcd_abs_eq
tff(fact_4155_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Max_singleton
tff(fact_4156_drop__upd__irrelevant,axiom,
    ! [A: $tType,M: nat,N2: nat,L: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
     => ( drop(A,N2,list_update(A,L,M,X)) = drop(A,N2,L) ) ) ).

% drop_upd_irrelevant
tff(fact_4157_drop__update__cancel,axiom,
    ! [A: $tType,N2: nat,M: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
     => ( drop(A,M,list_update(A,Xs,N2,X)) = drop(A,M,Xs) ) ) ).

% drop_update_cancel
tff(fact_4158_Gcd__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_sa(int,nat)),K5)) = aa(int,nat,nat2,gcd_Gcd(int,K5)) ).

% Gcd_nat_abs_eq
tff(fact_4159_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_il(nat,fun(nat,bool),N2))) = N2 ) ) ).

% Max_divisors_self_nat
tff(fact_4160_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_4161_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Max_less_iff
tff(fact_4162_Gcd__int__eq,axiom,
    ! [N7: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N7)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N7)) ).

% Gcd_int_eq
tff(fact_4163_Max__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A5: set(B),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( A5 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_sb(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% Max_const
tff(fact_4164_nth__drop,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,drop(A,N2,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),I2)) ) ) ).

% nth_drop
tff(fact_4165_set__drop__subset,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,N2,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_drop_subset
tff(fact_4166_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,N2,Xs)) ) ) ).

% sorted_drop
tff(fact_4167_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5))) ) ) ) ).

% Max_ge
tff(fact_4168_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ! [Y3: A] :
                ( pp(aa(set(A),bool,member(A,Y3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( pp(aa(set(A),bool,member(A,X),A5))
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_4169_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,finite_finite2(A),B4))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A5))
                 => ? [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),B4))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3)) ) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,member(A,X3),B4))
                   => ? [Xa3: A] :
                        ( pp(aa(set(A),bool,member(A,Xa3),A5))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3)) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,lattic643756798349783984er_Max(A),B4) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_4170_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,lattic643756798349783984er_Max(A),A5))) ) ) ) ).

% Max.coboundedI
tff(fact_4171_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5)),A5)) ) ) ) ).

% Max_in
tff(fact_4172_Gcd__int__greater__eq__0,axiom,
    ! [K5: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K5))) ).

% Gcd_int_greater_eq_0
tff(fact_4173_set__drop__subset__set__drop,axiom,
    ! [A: $tType,N2: nat,M: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,M,Xs))),aa(list(A),set(A),set2(A),drop(A,N2,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_4174_drop__update__swap,axiom,
    ! [A: $tType,M: nat,N2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( drop(A,M,list_update(A,Xs,N2,X)) = list_update(A,drop(A,M,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M),X) ) ) ).

% drop_update_swap
tff(fact_4175_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = M )
            <=> ( pp(aa(set(A),bool,member(A,M),A5))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M)) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4176_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_4177_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A5) )
            <=> ( pp(aa(set(A),bool,member(A,M),A5))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M)) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4178_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A14),X)) ) ) ) ) ) ).

% Max.boundedE
tff(fact_4179_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( pp(aa(set(A),bool,member(A,A6),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X)) ) ) ) ) ).

% Max.boundedI
tff(fact_4180_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_4181_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ! [B5: A] :
                ( pp(aa(set(A),bool,member(A,B5),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B5),A3)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = A3 ) ) ) ) ).

% Max_insert2
tff(fact_4182_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = aa(set(A),A,lattic643756798349783984er_Max(A),X6) ) ) ) ) ).

% cSup_eq_Max
tff(fact_4183_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).

% Max_Sup
tff(fact_4184_Sup__nat__def,axiom,
    ! [X6: set(nat)] :
      ( ( ( X6 = bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = zero_zero(nat) ) )
      & ( ( X6 != bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),X6) ) ) ) ).

% Sup_nat_def
tff(fact_4185_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N7: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M6),N7))
         => ( ( M6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N7))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M6)),aa(set(A),A,lattic643756798349783984er_Max(A),N7))) ) ) ) ) ).

% Max_mono
tff(fact_4186_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B4))) ) ) ) ) ).

% Max.subset_imp
tff(fact_4187_card__le__Suc__Max,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S)))) ) ).

% card_le_Suc_Max
tff(fact_4188_Max__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S: set(B),F3: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( S != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,A)),F3),K)),S)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,F3),S))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_4189_divide__nat__def,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( N2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = zero_zero(nat) ) )
      & ( ( N2 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N2) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_sd(nat,fun(nat,fun(nat,bool)),N2),M))) ) ) ) ).

% divide_nat_def
tff(fact_4190_in__set__drop__conv__nth,axiom,
    ! [A: $tType,X: A,N2: nat,L: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),drop(A,N2,L))))
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),I4))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),L)))
          & ( X = aa(nat,A,nth(A,L),I4) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_4191_Gcd__int__def,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),K5))) ).

% Gcd_int_def
tff(fact_4192_sum__le__card__Max,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image2(A,nat,F3),A5))))) ) ).

% sum_le_card_Max
tff(fact_4193_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_se(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_4194_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_4195_set__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set2(A),concat(A,Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_4196_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_4197_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_greaterThan(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),I2)) ) ) ).

% greaterThan_iff
tff(fact_4198_restrict__map__UNIV,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B))] : restrict_map(A,B,F3,top_top(set(A))) = F3 ).

% restrict_map_UNIV
tff(fact_4199_restrict__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),A5: set(A),B4: set(A)] : restrict_map(A,B,restrict_map(A,B,M,A5),B4) = restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ).

% restrict_restrict
tff(fact_4200_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D4: set(A),X5: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_bk(A,option(B)),D4),X5) = none(B) ).

% restrict_map_empty
tff(fact_4201_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),X)),aa(A,set(A),set_ord_greaterThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% greaterThan_subset_iff
tff(fact_4202_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),X5: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X5) = none(B) ).

% restrict_map_to_empty
tff(fact_4203_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,bool),set(int),collect(int),aTP_Lamp_ix(int,fun(int,bool),N2))) = aa(int,int,abs_abs(int),N2) ) ) ).

% Max_divisors_self_int
tff(fact_4204_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),top_top(A)))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_4205_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,X: A,D4: set(A),M: fun(A,option(B)),Y: option(B)] :
      ( pp(aa(set(A),bool,member(A,X),D4))
     => ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),X,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_4206_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,X: A,D4: set(A),M: fun(A,option(B)),Y: option(B)] :
      ( ( pp(aa(set(A),bool,member(A,X),D4))
       => ( restrict_map(A,B,fun_upd(A,option(B),M,X,Y),D4) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),X,Y) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),D4))
       => ( restrict_map(A,B,fun_upd(A,option(B),M,X,Y),D4) = restrict_map(A,B,M,D4) ) ) ) ).

% restrict_fun_upd
tff(fact_4207_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,X: A,D4: set(A),M: fun(A,option(B))] :
      ( ( pp(aa(set(A),bool,member(A,X),D4))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),D4))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X,none(B)) = restrict_map(A,B,M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_4208_restrict__map__eq_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A5: set(B),K: B,V2: A] :
      ( ( aa(B,option(A),restrict_map(B,A,M,A5),K) = aa(A,option(A),some(A),V2) )
    <=> ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V2) )
        & pp(aa(set(B),bool,member(B,K),A5)) ) ) ).

% restrict_map_eq(2)
tff(fact_4209_linorder_OMin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).

% linorder.Min.cong
tff(fact_4210_le__map__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M: fun(A,option(B)),X6: set(A)] : pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M,X6)),M)) ) ).

% le_map_restrict
tff(fact_4211_restrict__map__subset__eq,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),R2: set(A),M8: fun(A,option(B)),R8: set(A)] :
      ( ( restrict_map(A,B,M,R2) = M8 )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R8),R2))
       => ( restrict_map(A,B,M,R8) = restrict_map(A,B,M8,R8) ) ) ) ).

% restrict_map_subset_eq
tff(fact_4212_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] : aa(A,set(A),set_ord_greaterThan(A),X) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_4213_foldl__foldl__conv__concat,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,A)),A3: A,Xs: list(list(B))] : aa(list(list(B)),A,aa(A,fun(list(list(B)),A),foldl(A,list(B),foldl(A,B,F3)),A3),Xs) = aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),A3),concat(B,Xs)) ).

% foldl_foldl_conv_concat
tff(fact_4214_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_4215_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y: A,M: fun(B,option(A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,Y),ran(B,A,restrict_map(B,A,M,A5))))
     => ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A5))
          & ( aa(B,option(A),M,X3) = aa(A,option(A),some(A),Y) ) ) ) ).

% ran_restrictD
tff(fact_4216_map__restrict__insert__none__simp,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),X: B,S2: set(B)] :
      ( ( aa(B,option(A),M,X) = none(A) )
     => ( restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),S2))) = restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),S2)) ) ) ).

% map_restrict_insert_none_simp
tff(fact_4217_restrict__map__upd,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),S: set(A),K: A,V2: B] : fun_upd(A,option(B),restrict_map(A,B,F3,S),K,aa(B,option(B),some(B),V2)) = restrict_map(A,B,fun_upd(A,option(B),F3,K,aa(B,option(B),some(B),V2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),S)) ).

% restrict_map_upd
tff(fact_4218_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_4219_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),D4: set(A),X: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),X,Y) ).

% fun_upd_restrict
tff(fact_4220_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X: A] : restrict_map(A,B,F3,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = fun_upd(A,option(B),F3,X,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_4221_map__upd__eq__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A] : fun_upd(A,option(B),M,X,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ).

% map_upd_eq_restrict
tff(fact_4222_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( pp(aa(set(list(A)),bool,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs2: list(A)] :
              ( pp(aa(set(list(A)),bool,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
             => ( pp(aa(set(list(A)),bool,member(list(A),Zs2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
               => ( ( Ys3 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_4223_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_4224_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ).

% length_remdups_concat
tff(fact_4225_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),D4: set(A),M: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D4))
       => ( restrict_map(A,B,map_upds(A,B,M,Xs,Ys2),D4) = map_upds(A,B,restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(list(A),set(A),set2(A),Xs))),Xs,Ys2) ) ) ) ).

% restrict_map_upds
tff(fact_4226_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),aa(list(list(A)),list(list(A)),removeAll(list(A),nil(A)),Xs))
        & ! [Ys4: list(A)] :
            ( pp(aa(set(list(A)),bool,member(list(A),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( pp(aa(set(list(A)),bool,member(list(A),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
              & pp(aa(set(list(A)),bool,member(list(A),Zs3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
              & ( Ys4 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys4)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_4227_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F3),top_top(set(nat)))))
         => ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
           => ( ! [N5: nat] :
                  ( ( aa(nat,A,F3,N5) = aa(nat,A,F3,aa(nat,nat,suc,N5)) )
                 => ( aa(nat,A,F3,aa(nat,nat,suc,N5)) = aa(nat,A,F3,aa(nat,nat,suc,aa(nat,nat,suc,N5))) ) )
             => ? [N11: nat] :
                  ( ! [N8: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N11))
                     => ! [M4: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N11))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N8))
                           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,M4)),aa(nat,A,F3,N8))) ) ) )
                  & ! [N8: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N11),N8))
                     => ( aa(nat,A,F3,N11) = aa(nat,A,F3,N8) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_4228_slice__eq__bounds__empty,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : slice(A,I2,I2,Xs) = nil(A) ).

% slice_eq_bounds_empty
tff(fact_4229_slice__Nil,axiom,
    ! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).

% slice_Nil
tff(fact_4230_set__empty,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty
tff(fact_4231_set__empty2,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty2
tff(fact_4232_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_4233_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_remdups_leq
tff(fact_4234_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_4235_drop__eq__Nil2,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,N2,Xs) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N2)) ) ).

% drop_eq_Nil2
tff(fact_4236_drop__eq__Nil,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( ( drop(A,N2,Xs) = nil(A) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N2)) ) ).

% drop_eq_Nil
tff(fact_4237_drop__all,axiom,
    ! [A: $tType,Xs: list(A),N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N2))
     => ( drop(A,N2,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_4238_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = nil(A) )
          <=> ( A5 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4239_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,M: fun(A,option(B)),Ys2: list(B),Y: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( map_upds(A,B,M,Xs,list_update(B,Ys2,I2,Y)) = map_upds(A,B,M,Xs,Ys2) ) ) ).

% map_upds_list_update2_drop
tff(fact_4240_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A3: A,As: list(A),M: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),aa(list(A),set(A),set2(A),As)))
     => ( map_upds(A,B,fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),B2)),As,Bs) = fun_upd(A,option(B),map_upds(A,B,M,As,Bs),A3,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_4241_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L)))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_4242_merge_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L22: list(A)] : merge(A,nil(A),L22) = L22 ) ).

% merge.simps(1)
tff(fact_4243_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ys2: list(A)] : shuffles(A,nil(A),Ys2) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Ys2),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_4244_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xs),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_4245_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),aTP_Lamp_sf(fun(A,A),fun(nat,A),Q))) ) ) ).

% mono_funpow
tff(fact_4246_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3))) ) ) ).

% mono_pow
tff(fact_4247_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoD
tff(fact_4248_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoE
tff(fact_4249_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
         => pp(aa(fun(A,B),bool,order_mono(A,B),F3)) ) ) ).

% monoI
tff(fact_4250_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y5))) ) ) ) ).

% mono_def
tff(fact_4251_mono__Suc,axiom,
    pp(aa(fun(nat,nat),bool,order_mono(nat,nat),suc)) ).

% mono_Suc
tff(fact_4252_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mono_strict_invE
tff(fact_4253_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mono_invE
tff(fact_4254_mono__iff__le__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
        <=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N)))) ) ) ).

% mono_iff_le_Suc
tff(fact_4255_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F3: fun(A,B),A5: A,B4: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B4))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,A5)),aa(A,B,F3,B4)))) ) ) ).

% mono_inf
tff(fact_4256_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F3: fun(A,B),A5: A,B4: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,A5)),aa(A,B,F3,B4))),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B4)))) ) ) ).

% mono_sup
tff(fact_4257_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),A5: A,B4: A,N2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),A5)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),B4))) ) ) ) ).

% funpow_mono
tff(fact_4258_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_4259_len__greater__imp__nonempty,axiom,
    ! [A: $tType,X: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),L)))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_4260_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_4261_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_4262_sorted__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),remdups(A),Xs)) ) ) ).

% sorted_remdups
tff(fact_4263_Rings_Omono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),times_times(A),A3))) ) ) ).

% Rings.mono_mult
tff(fact_4264_mono__times__nat,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),N2))) ) ).

% mono_times_nat
tff(fact_4265_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F3: fun(A,B),M: A,N2: A,M8: B,N6: B] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( ( aa(set(A),set(B),image2(A,B,F3),set_or7035219750837199246ssThan(A,M,N2)) = set_or7035219750837199246ssThan(B,M8,N6) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N2))
             => ( aa(A,B,F3,M) = M8 ) ) ) ) ) ).

% mono_image_least
tff(fact_4266_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F3: fun(A,A),P3: A,K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,P3)),P3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F3),bot_bot(A))),P3)) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_4267_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F3: fun(A,A),P3: A,K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,F3,P3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F3),top_top(A)))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_4268_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),I2: nat,J: nat,X: A,Y: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F3,X)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I2),F3),X)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F3),Y))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_4269_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A5)))) ) ) ).

% mono_Sup
tff(fact_4270_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sg(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A5),I5))))) ) ) ).

% mono_SUP
tff(fact_4271_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F3: fun(A,B),A5: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A5),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sg(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I5)))) ) ) ).

% mono_INF
tff(fact_4272_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5)))) ) ) ).

% mono_Inf
tff(fact_4273_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)) = aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),L)) ).

% length_remdups_card
tff(fact_4274_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,N2: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),bot_bot(A)))) ) ) ) ).

% funpow_decreasing
tff(fact_4275_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,N2: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_4276_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F3),A5)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_4277_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aTP_Lamp_sh(nat,fun(nat,nat),K))) ) ).

% mono_ge2_power_minus_self
tff(fact_4278_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys2 )
    <=> ? [F10: fun(nat,nat)] :
          ( pp(aa(fun(nat,nat),bool,order_mono(nat,nat),F10))
          & ( aa(set(nat),set(nat),image2(nat,nat,F10),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys2)) )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys2),aa(nat,nat,F10,I4)) ) )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) )
              <=> ( aa(nat,nat,F10,I4) = aa(nat,nat,F10,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_4279_distinct__concat_H,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_si(list(A),bool)),Xs))
     => ( ! [Ys3: list(A)] :
            ( pp(aa(set(list(A)),bool,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs2: list(A)] :
              ( pp(aa(set(list(A)),bool,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
             => ( pp(aa(set(list(A)),bool,member(list(A),Zs2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
               => ( ( Ys3 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat'
tff(fact_4280_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B4: set(A),X: B,A5: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( 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_sj(B,fun(A,set(product_prod(B,A))),X)),B4))),A5) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_sk(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A5,B4) ) ) ).

% fold_union_pair
tff(fact_4281_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(set(nat),set(A),image2(nat,A,nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),take(A,L,Xs)) ) ) ).

% nth_image
tff(fact_4282_filter__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),filter2(A,Q),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_sl(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)),Xs) ).

% filter_filter
tff(fact_4283_fold__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Z2: A] : finite_fold(B,A,F3,Z2,bot_bot(set(B))) = Z2 ).

% fold_empty
tff(fact_4284_take__update,axiom,
    ! [A: $tType,N2: nat,L: list(A),I2: nat,X: A] : take(A,N2,list_update(A,L,I2,X)) = list_update(A,take(A,N2,L),I2,X) ).

% take_update
tff(fact_4285_set__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs)) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_sm(fun(A,bool),fun(list(A),fun(A,bool)),P),Xs)) ).

% set_filter
tff(fact_4286_take__all,axiom,
    ! [A: $tType,Xs: list(A),N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N2))
     => ( take(A,N2,Xs) = Xs ) ) ).

% take_all
tff(fact_4287_take__all__iff,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( ( take(A,N2,Xs) = Xs )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N2)) ) ).

% take_all_iff
tff(fact_4288_nth__take,axiom,
    ! [A: $tType,I2: nat,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N2))
     => ( aa(nat,A,nth(A,take(A,N2,Xs)),I2) = aa(nat,A,nth(A,Xs),I2) ) ) ).

% nth_take
tff(fact_4289_take__update__cancel,axiom,
    ! [A: $tType,N2: nat,M: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => ( take(A,N2,list_update(A,Xs,M,Y)) = take(A,N2,Xs) ) ) ).

% take_update_cancel
tff(fact_4290_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] : pp(aa(set(list(A)),bool,member(list(A),Xs),shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,aTP_Lamp_ci(fun(A,bool),fun(A,bool),P)),Xs)))) ).

% partition_in_shuffles
tff(fact_4291_removeAll__filter__not__eq,axiom,
    ! [A: $tType,X: A] : removeAll(A,X) = filter2(A,aa(A,fun(A,bool),aTP_Lamp_sn(A,fun(A,bool)),X)) ).

% removeAll_filter_not_eq
tff(fact_4292_filter__nth__ex__nth,axiom,
    ! [A: $tType,N2: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))))
     => ? [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M5))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,aa(list(A),list(A),filter2(A,P),Xs)),N2) = aa(nat,A,nth(A,Xs),M5) )
          & ( aa(list(A),list(A),filter2(A,P),take(A,M5,Xs)) = take(A,N2,aa(list(A),list(A),filter2(A,P),Xs)) ) ) ) ).

% filter_nth_ex_nth
tff(fact_4293_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% filter_is_subset
tff(fact_4294_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_filter_le
tff(fact_4295_sorted__filter_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,P),L)) ) ) ).

% sorted_filter'
tff(fact_4296_mono__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(set(A),set(B)),A5: set(A),B4: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F3,A5)),aa(set(A),set(B),F3,B4)))) ) ).

% mono_Int
tff(fact_4297_mono__Un,axiom,
    ! [B: $tType,A: $tType,F3: fun(set(A),set(B)),A5: set(A),B4: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),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),F3,A5)),aa(set(A),set(B),F3,B4))),aa(set(A),set(B),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))) ) ).

% mono_Un
tff(fact_4298_set__take__subset,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,N2,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_take_subset
tff(fact_4299_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,N2,Xs)) ) ) ).

% sorted_take
tff(fact_4300_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% remdups_adj_length
tff(fact_4301_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_ci(fun(A,bool),fun(A,bool),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_4302_inter__set__filter,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)),Xs)) ).

% inter_set_filter
tff(fact_4303_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G3: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,bool),aTP_Lamp_so(fun(list(A),A),fun(list(A),fun(A,bool)),G3),Xs)),Xs)) ) ).

% sorted_same
tff(fact_4304_sorted__remdups__adj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups_adj(A,Xs)) ) ) ).

% sorted_remdups_adj
tff(fact_4305_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_si(list(A),bool)),Xs)) = concat(A,Xs) ).

% concat_filter_neq_Nil
tff(fact_4306_length__filter__less,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% length_filter_less
tff(fact_4307_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,M,Xs))),aa(list(A),set(A),set2(A),take(A,N2,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_4308_union__fold__insert,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4) = finite_fold(A,set(A),insert(A),B4,A5) ) ) ).

% union_fold_insert
tff(fact_4309_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B4) = finite_fold(A,A,sup_sup(A),B4,A5) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_4310_inf__Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B4: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A5)),B4) = finite_fold(A,A,inf_inf(A),B4,A5) ) ) ) ).

% inf_Inf_fold_inf
tff(fact_4311_slice__def,axiom,
    ! [A: $tType,From: nat,To: nat,List: list(A)] : slice(A,From,To,List) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From),drop(A,From,List)) ).

% slice_def
tff(fact_4312_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys2: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys2)))
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys2),I3) ) )
         => ( take(A,K,Xs) = take(A,K,Ys2) ) ) ) ) ).

% nth_take_lemma
tff(fact_4313_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),aTP_Lamp_sp(A,fun(A,bool)),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_4314_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,complete_Sup_Sup(A),A5) = finite_fold(A,A,sup_sup(A),bot_bot(A),A5) ) ) ) ).

% Sup_fold_sup
tff(fact_4315_Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,complete_Inf_Inf(A),A5) = finite_fold(A,A,inf_inf(A),top_top(A),A5) ) ) ) ).

% Inf_fold_inf
tff(fact_4316_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_sq(list(A),fun(A,bool),Xs)),Zs) = Ys2 ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_4317_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_sr(list(A),fun(A,bool),Xs)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_4318_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_sq(list(A),fun(A,bool),Ys2)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_4319_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_sr(list(A),fun(A,bool),Ys2)),Zs) = Ys2 ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_4320_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P3),Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_ss(fun(A,bool),fun(list(A),fun(nat,bool)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_4321_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G3),one_one(A),A5) ) ).

% prod.eq_fold
tff(fact_4322_remdups__adj__adjacent,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I2) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I2)) ) ) ).

% remdups_adj_adjacent
tff(fact_4323_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(A),set(B),image2(A,B,F3),A5) = finite_fold(A,set(B),aTP_Lamp_st(fun(A,B),fun(A,fun(set(B),set(B))),F3),bot_bot(set(B)),A5) ) ) ).

% image_fold_insert
tff(fact_4324_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_4325_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,X: A,Ys2: list(B),Xs: list(A),F3: fun(A,option(B)),Y: B] :
      ( ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys2) = map_upds(A,B,F3,Xs,Ys2) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys2) = fun_upd(A,option(B),map_upds(A,B,F3,Xs,Ys2),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_4326_Union__take__drop__id,axiom,
    ! [A: $tType,N2: nat,L: list(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),N2,L)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),take(set(A),N2,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_4327_sup__SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: A,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),B4),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F3),B4,A5) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_4328_inf__INF__fold__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: A,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),B4),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F3),B4,A5) ) ) ) ).

% inf_INF_fold_inf
tff(fact_4329_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))) ) ).

% remdups_adj_length_ge1
tff(fact_4330_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I2: nat,J: nat] :
      ( distinct(A,Vs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I2,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_4331_foldl__list__update,axiom,
    ! [B: $tType,A: $tType,N2: nat,Xs: list(A),F3: fun(B,fun(A,B)),A3: B,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),A3),list_update(A,Xs,N2,X)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),aa(A,B,aa(B,fun(A,B),F3,aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),A3),take(A,N2,Xs))),X)),drop(A,aa(nat,nat,suc,N2),Xs)) ) ) ).

% foldl_list_update
tff(fact_4332_SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F3),bot_bot(A),A5) ) ) ) ).

% SUP_fold_sup
tff(fact_4333_INF__fold__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F3),top_top(A),A5) ) ) ) ).

% INF_fold_inf
tff(fact_4334_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_4335_Set__filter__fold,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( filter3(A,P,A5) = finite_fold(A,set(A),aTP_Lamp_su(fun(A,bool),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A5) ) ) ).

% Set_filter_fold
tff(fact_4336_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V2: B] : graph(A,B,fun_upd(A,option(B),M,K,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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).

% graph_map_upd
tff(fact_4337_shuffles_Opsimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))))
     => ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_4338_member__filter,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),filter3(A,P,A5)))
    <=> ( pp(aa(set(A),bool,member(A,X),A5))
        & pp(aa(A,bool,P,X)) ) ) ).

% member_filter
tff(fact_4339_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_bk(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_4340_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,bool),A5: set(A)] : filter3(A,P,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_sv(fun(A,bool),fun(set(A),fun(A,bool)),P),A5)) ).

% Set.filter_def
tff(fact_4341_in__graphD,axiom,
    ! [A: $tType,B: $tType,K: A,V2: B,M: fun(A,option(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,M)))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V2) ) ) ).

% in_graphD
tff(fact_4342_in__graphI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V2: A] :
      ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V2) )
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V2)),graph(B,A,M))) ) ).

% in_graphI
tff(fact_4343_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V2: B,M: fun(A,option(B)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,restrict_map(A,B,M,A5))))
     => pp(aa(set(A),bool,member(A,K),A5)) ) ).

% graph_restrictD(1)
tff(fact_4344_card_Oeq__fold,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = finite_fold(A,nat,aTP_Lamp_sw(A,fun(nat,nat)),zero_zero(nat),A5) ).

% card.eq_fold
tff(fact_4345_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),nil(A),A5) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_4346_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K: A,V2: B,M: fun(A,option(B)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,restrict_map(A,B,M,A5))))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V2) ) ) ).

% graph_restrictD(2)
tff(fact_4347_inter__Set__filter,axiom,
    ! [A: $tType,B4: set(A),A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = filter3(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5),B4) ) ) ).

% inter_Set_filter
tff(fact_4348_shuffles_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys2: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)))
     => ( shuffles(A,nil(A),Ys2) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Ys2),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_4349_coinduct3__mono__lemma,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [F3: fun(A,set(B)),X6: set(B),B4: set(B)] :
          ( pp(aa(fun(A,set(B)),bool,order_mono(A,set(B)),F3))
         => pp(aa(fun(A,set(B)),bool,order_mono(A,set(B)),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_sx(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),F3),X6),B4))) ) ) ).

% coinduct3_mono_lemma
tff(fact_4350_Pow__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pow(A,A5) = finite_fold(A,set(set(A)),aTP_Lamp_sy(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A5) ) ) ).

% Pow_fold
tff(fact_4351_shuffles_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa2)))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa2),bot_bot(set(list(A)))) )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,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)),Xa2))) ) )
         => ( ( ( Xa2 = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) )
               => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A)))) ) )
           => ~ ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ! [Y3: A,Ys3: list(A)] :
                      ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                     => ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys3))) )
                       => ~ pp(aa(product_prod(list(A),list(A)),bool,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),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_4352_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lex(A,R3)))
     => ~ ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys2)))
             => ( ( take(A,I3,Xs) = take(A,I3,Ys2) )
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Ys2),I3))),R3)) ) ) ) ) ).

% lex_take_index
tff(fact_4353_Pow__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(set(A)),bool,member(set(A),A5),pow(A,B4)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ).

% Pow_iff
tff(fact_4354_PowI,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(set(A)),bool,member(set(A),A5),pow(A,B4))) ) ).

% PowI
tff(fact_4355_Pow__UNIV,axiom,
    ! [A: $tType] : pow(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_4356_Pow__Int__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) = 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,A5)),pow(A,B4)) ).

% Pow_Int_eq
tff(fact_4357_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,X: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_4358_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N2: nat,X: B,Xs: list(B)] : enumerate(B,N2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),N2),X)),enumerate(B,aa(nat,nat,suc,N2),Xs)) ).

% enumerate_simps(2)
tff(fact_4359_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,As: list(A),B2: B,Bs: list(B)] : map_upds(A,B,M,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),B2)),As,Bs) ).

% map_upds_Cons
tff(fact_4360_Pow__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A)] :
      ( ( pow(A,X6) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Y6),bot_bot(set(set(A)))) )
    <=> ( ( X6 = bot_bot(set(A)) )
        & ( Y6 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_4361_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))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_4362_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),lex(A,R3)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lex(A,R3))) ) ) ) ).

% Cons_in_lex
tff(fact_4363_nth__Cons__pos,axiom,
    ! [A: $tType,N2: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_4364_neq__NilE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [X3: A,Xs2: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) ) ).

% neq_NilE
tff(fact_4365_list__2pre__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),W1: list(A),W22: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
     => ( ! [E2: A,W12: list(A),W23: list(B)] :
            ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W12),W23))
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,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)] :
              ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W13),W24))
             => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),W24))) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W1),W22)) ) ) ) ).

% list_2pre_induct
tff(fact_4366_list__induct__first2,axiom,
    ! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
      ( pp(aa(list(A),bool,P,nil(A)))
     => ( ! [X3: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))
       => ( ! [X12: A,X22: A,Xs2: list(A)] :
              ( pp(aa(list(A),bool,P,Xs2))
             => pp(aa(list(A),bool,P,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),X22),Xs2)))) )
         => pp(aa(list(A),bool,P,Xs)) ) ) ) ).

% list_induct_first2
tff(fact_4367_mergesort__by__rel__merge__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),R2: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( ! [Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),nil(B)))
     => ( ! [Ys3: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),Ys3))
       => ( ! [X3: A,Xs2: list(A),Y3: B,Ys3: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Y3))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3)))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3))) ) )
         => ( ! [X3: A,Xs2: list(A),Y3: B,Ys3: list(B)] :
                ( ~ pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Y3))
               => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3))
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3))) ) )
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys2)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4368_successively_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P2: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),nil(A))
     => ( ! [P2: fun(A,fun(A,bool)),X3: A] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ~ ! [P2: fun(A,fun(A,bool)),X3: A,Y3: A,Xs2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs2))) ) ) ).

% successively.cases
tff(fact_4369_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: product_prod(fun(A,B),list(A))] :
          ( ! [F2: fun(A,B),X3: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
         => ( ! [F2: fun(A,B),X3: A,Y3: A,Zs2: list(A)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))
           => ~ ! [A6: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A6),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_4370_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P2: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),nil(A))
     => ~ ! [P2: fun(A,fun(A,bool)),X3: A,Ys3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3)) ) ).

% sorted_wrt.cases
tff(fact_4371_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xs: list(A),Ys2: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,Xs,Ys2))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))) ).

% Cons_shuffles_subset2
tff(fact_4372_Cons__shuffles__subset1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),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),X)),shuffles(A,Xs,Ys2))),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2))) ).

% Cons_shuffles_subset1
tff(fact_4373_PowD,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(set(A)),bool,member(set(A),A5),pow(A,B4)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ).

% PowD
tff(fact_4374_Pow__top,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(set(A)),bool,member(set(A),A5),pow(A,A5))) ).

% Pow_top
tff(fact_4375_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_4376_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2))) ).

% shuffles.simps(3)
tff(fact_4377_Pow__bottom,axiom,
    ! [A: $tType,B4: set(A)] : pp(aa(set(set(A)),bool,member(set(A),bot_bot(set(A))),pow(A,B4))) ).

% Pow_bottom
tff(fact_4378_Pow__not__empty,axiom,
    ! [A: $tType,A5: set(A)] : pow(A,A5) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_4379_Pow__def,axiom,
    ! [A: $tType,A5: set(A)] : pow(A,A5) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ie(set(A),fun(set(A),bool),A5)) ).

% Pow_def
tff(fact_4380_drop__Cons,axiom,
    ! [A: $tType,N2: nat,X: A,Xs: list(A)] : drop(A,N2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aTP_Lamp_sz(list(A),fun(nat,list(A)),Xs),N2) ).

% drop_Cons
tff(fact_4381_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I2: nat,V2: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),I2,V2) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_ta(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V2),I2) ).

% list_update.simps(2)
tff(fact_4382_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) ).

% set_subset_Cons
tff(fact_4383_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2) ) ) ).

% impossible_Cons
tff(fact_4384_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_4385_subset__eq__mset__impl_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)
     => ~ ! [X3: A,Xs2: list(A),Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3) ) ).

% subset_eq_mset_impl.cases
tff(fact_4386_zipf_Ocases,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,fun(B,C))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [F2: fun(A,fun(B,C)),A6: A,As4: list(A),B5: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))
       => ( ! [A6: fun(A,fun(B,C)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A6),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))
         => ~ ! [A6: fun(A,fun(B,C)),V3: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A6),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))) ) ) ) ).

% zipf.cases
tff(fact_4387_merge_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(A),list(A))] :
          ( ! [L23: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),L23)
         => ( ! [V3: A,Va: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(A))
           => ~ ! [X12: A,L12: list(A),X22: A,L23: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23)) ) ) ) ).

% merge.cases
tff(fact_4388_shuffles_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)
     => ( ! [Xs2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A))
       => ~ ! [X3: A,Xs2: list(A),Y3: A,Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)) ) ) ).

% shuffles.cases
tff(fact_4389_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))] :
      ( ! [P2: fun(A,fun(B,bool))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [P2: fun(A,fun(B,bool)),A6: A,As4: list(A),B5: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))
       => ( ! [P2: fun(A,fun(B,bool)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))
         => ~ ! [P2: fun(A,fun(B,bool)),V3: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_4390_partition__rev_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))] :
      ( ! [P2: fun(A,bool),Yes: list(A),No: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P2),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
     => ~ ! [P2: fun(A,bool),Yes: list(A),No: list(A),X3: A,Xs2: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P2),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ).

% partition_rev.cases
tff(fact_4391_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,B),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs2))
     => ~ ! [F2: fun(A,B),A6: A,As4: list(A),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_4392_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
      ( ! [R9: fun(A,fun(A,bool)),Sl: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R9),aa(list(A),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)))
     => ~ ! [R9: fun(A,fun(A,bool)),Sl: list(A),X3: A,Xs2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R9),aa(list(A),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),X3),Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_4393_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
      ( ! [R9: fun(A,fun(A,bool)),X3: A,Xs2: list(A),Y3: A,Ys3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R9),aa(list(A),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),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))
     => ( ! [R9: fun(A,fun(A,bool)),Xs2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R9),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A)))
       => ~ ! [R9: fun(A,fun(A,bool)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R9),aa(list(A),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_4394_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,X: product_prod(product_prod(list(A),list(A)),list(A))] :
      ( ! [Xs1: list(A),Xs22: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),nil(A))
     => ( ! [Xs1: list(A),Xs22: list(A),X3: A] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ~ ! [Xs1: list(A),Xs22: list(A),X12: A,X22: A,Xs2: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),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),X22),Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_4395_drop__eq__ConsD,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),X: A,Xs4: list(A)] :
      ( ( drop(A,N2,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs4) )
     => ( drop(A,aa(nat,nat,suc,N2),Xs) = Xs4 ) ) ).

% drop_eq_ConsD
tff(fact_4396_insort__key_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Y: B,Ys2: list(B)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Ys2)) ) ) ) ) ).

% insort_key.simps(2)
tff(fact_4397_merge_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X1: A,X2: A,L1: list(A),L22: list(A)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X2))
           => ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X2))
           => ( ( ( X1 = X2 )
               => ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,L22)) ) )
              & ( ( X1 != X2 )
               => ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),L22)) ) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_4398_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_4399_Pow__mono,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pow(A,A5)),pow(A,B4))) ) ).

% Pow_mono
tff(fact_4400_subset__Pow__Union,axiom,
    ! [A: $tType,A5: set(set(A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A5),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)))) ).

% subset_Pow_Union
tff(fact_4401_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B4: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A5) = B4 )
     => ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),pow(B,A5)) = pow(A,B4) ) ) ).

% image_Pow_surj
tff(fact_4402_Pow__set_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : pow(B,aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs))) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),pow(B,aa(list(B),set(B),set2(B),Xs))),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X)),pow(B,aa(list(B),set(B),set2(B),Xs)))) ).

% Pow_set(2)
tff(fact_4403_take__Cons,axiom,
    ! [A: $tType,N2: nat,X: A,Xs: list(A)] : take(A,N2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_tb(A,fun(list(A),fun(nat,list(A))),X),Xs),N2) ).

% take_Cons
tff(fact_4404_Pow__INT__eq,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B)] : pow(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))) = aa(set(set(set(A))),set(set(A)),complete_Inf_Inf(set(set(A))),aa(set(B),set(set(set(A))),image2(B,set(set(A)),aTP_Lamp_tc(fun(B,set(A)),fun(B,set(set(A))),B4)),A5)) ).

% Pow_INT_eq
tff(fact_4405_nth__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),N2: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N2) = case_nat(A,X,nth(A,Xs),N2) ).

% nth_Cons
tff(fact_4406_Fpow__subset__Pow,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A5)),pow(A,A5))) ).

% Fpow_subset_Pow
tff(fact_4407_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))),lex(A,R3))) ).

% Nil2_notin_lex
tff(fact_4408_Nil__notin__lex,axiom,
    ! [A: $tType,Ys2: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)),lex(A,R3))) ).

% Nil_notin_lex
tff(fact_4409_length__compl__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),L: list(A)] :
      ( pp(aa(list(A),bool,P,nil(A)))
     => ( ! [E2: A,L4: list(A)] :
            ( ! [Ll: list(A)] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L4)))
               => pp(aa(list(A),bool,P,Ll)) )
           => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L4))) )
       => pp(aa(list(A),bool,P,L)) ) ) ).

% length_compl_induct
tff(fact_4410_Suc__le__length__iff,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N2)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ? [X4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_4411_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).

% sorted1
tff(fact_4412_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Ys2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) )
            & sorted_wrt(A,ord_less_eq(A),Ys2) ) ) ) ).

% sorted_simps(2)
tff(fact_4413_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A6: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),nil(A)) ) ).

% list_decomp_1
tff(fact_4414_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Ys2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) )
            & sorted_wrt(A,ord_less(A),Ys2) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_4415_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X6: fun(bool,A),Y6: fun(bool,A)] :
          ( pp(aa(fun(bool,A),bool,aa(fun(bool,A),fun(fun(bool,A),bool),ord_less_eq(fun(bool,A)),X6),Y6))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fFalse)),aa(bool,A,Y6,fFalse)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fTrue)),aa(bool,A,Y6,fTrue))) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_4416_insort__is__Cons,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),F3: fun(B,A),A3: B] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),aa(list(B),set(B),set2(B),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,A3)),aa(B,A,F3,X3))) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),A3),Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_4417_merge_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Xa2: list(A),Y: list(A)] :
          ( ( merge(A,X,Xa2) = Y )
         => ( ( ( X = nil(A) )
             => ( Y != Xa2 ) )
           => ( ! [V3: A,Va: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( ( Xa2 = nil(A) )
                   => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) ) ) )
             => ~ ! [X12: A,L12: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12) )
                   => ! [X22: A,L23: list(A)] :
                        ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23) )
                       => ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                             => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23))) ) )
                            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                             => ( ( ( X12 = X22 )
                                 => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,L23)) ) )
                                & ( ( X12 != X22 )
                                 => ( Y = 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),X12),L12),L23)) ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
tff(fact_4418_Un__Pow__subset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),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,A5)),pow(A,B4))),pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))) ).

% Un_Pow_subset
tff(fact_4419_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image2(B,set(set(A)),aTP_Lamp_tc(fun(B,set(A)),fun(B,set(set(A))),B4)),A5))),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5))))) ).

% UN_Pow_subset
tff(fact_4420_Pow__insert,axiom,
    ! [A: $tType,A3: A,A5: set(A)] : pow(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = 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,A5)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3)),pow(A,A5))) ).

% Pow_insert
tff(fact_4421_Fpow__Pow__finite,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) = 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,A5)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_4422_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A5)),B4))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),pow(B,A5))),pow(A,B4))) ) ).

% image_Pow_mono
tff(fact_4423_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
      ( pp(aa(product_prod(list(A),list(A)),bool,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)))
     => ( ! [Ys3: list(A)] :
            ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
       => ( ! [Xs2: list(A)] :
              ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A))))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),nil(A))) )
         => ( ! [X3: A,Xs2: list(A),Y3: A,Ys3: list(A)] :
                ( pp(aa(product_prod(list(A),list(A)),bool,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),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))))
               => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))
                 => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3))
                   => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))) ) ) )
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ) ).

% shuffles.pinduct
tff(fact_4424_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))))
     => ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2))) ) ) ).

% shuffles.psimps(3)
tff(fact_4425_shuffles_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa2),bot_bot(set(list(A)))) ) )
       => ( ( ( Xa2 = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ! [Y3: A,Ys3: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                   => ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys3))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_4426_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
     => ? [A6: A,B5: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B5),nil(A))) ) ).

% list_decomp_2
tff(fact_4427_binomial__def,axiom,
    ! [N2: nat,K: nat] : aa(nat,nat,binomial(N2),K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),bool),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),bool),aTP_Lamp_td(nat,fun(nat,fun(set(nat),bool)),N2),K))) ).

% binomial_def
tff(fact_4428_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N2: nat] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N2) = X )
        <=> ( N2 = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4429_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A),N2: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N2) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) = Y )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4430_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs)) = drop(A,I2,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_4431_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,X: A,Xs: list(A)] :
      ( ( ( ( Begin = zero_zero(nat) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
       => ( slice(A,Begin,End,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),slice(A,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero(nat) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
       => ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Begin),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs) ) ) ) ).

% slice_Cons
tff(fact_4432_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))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_4433_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A5: set(A)] : set_Cons(A,A5,aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_te(A,list(A))),A5) ).

% set_Cons_sing_Nil
tff(fact_4434_merge_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Xa2: list(A),Y: list(A)] :
          ( ( merge(A,X,Xa2) = Y )
         => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa2)))
           => ( ( ( X = nil(A) )
               => ( ( Y = Xa2 )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,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)),Xa2))) ) )
             => ( ! [V3: A,Va: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                   => ( ( Xa2 = nil(A) )
                     => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                       => ~ pp(aa(product_prod(list(A),list(A)),bool,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,L12: list(A)] :
                      ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12) )
                     => ! [X22: A,L23: list(A)] :
                          ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23) )
                         => ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                               => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23))) ) )
                              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                               => ( ( ( X12 = X22 )
                                   => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,L23)) ) )
                                  & ( ( X12 != X22 )
                                   => ( Y = 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),X12),L12),L23)) ) ) ) ) )
                           => ~ pp(aa(product_prod(list(A),list(A)),bool,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),L12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23)))) ) ) ) ) ) ) ) ) ).

% merge.pelims
tff(fact_4435_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),M: fun(A,option(B)),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2)))
     => ( map_upds(A,B,M,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))),Ys2) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys2),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_4436_empty__append__eq__id,axiom,
    ! [A: $tType,X5: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),X5) = X5 ).

% empty_append_eq_id
tff(fact_4437_list__ee__eq__leel_I1_J,axiom,
    ! [A: $tType,E1: A,E22: A,L1: list(A),E12: A,E23: A,L22: 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),L1),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),L22))) )
    <=> ( ( L1 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L22 = nil(A) ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_4438_list__ee__eq__leel_I2_J,axiom,
    ! [A: $tType,L1: list(A),E12: A,E23: A,L22: list(A),E1: A,E22: A] :
      ( ( 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),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L22))) = 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))) )
    <=> ( ( L1 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L22 = nil(A) ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_4439_list__se__match_I1_J,axiom,
    ! [A: $tType,L1: list(A),L22: list(A),A3: A] :
      ( ( L1 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L22) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
      <=> ( ( L1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
          & ( L22 = nil(A) ) ) ) ) ).

% list_se_match(1)
tff(fact_4440_list__se__match_I2_J,axiom,
    ! [A: $tType,L22: list(A),L1: list(A),A3: A] :
      ( ( L22 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L22) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
      <=> ( ( L1 = nil(A) )
          & ( L22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).

% list_se_match(2)
tff(fact_4441_list__se__match_I3_J,axiom,
    ! [A: $tType,L1: list(A),A3: A,L22: list(A)] :
      ( ( L1 != 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),L1),L22) )
      <=> ( ( L1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
          & ( L22 = nil(A) ) ) ) ) ).

% list_se_match(3)
tff(fact_4442_list__se__match_I4_J,axiom,
    ! [A: $tType,L22: list(A),A3: A,L1: list(A)] :
      ( ( L22 != 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),L1),L22) )
      <=> ( ( L1 = nil(A) )
          & ( L22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).

% list_se_match(4)
tff(fact_4443_list__e__eq__lel_I1_J,axiom,
    ! [A: $tType,E3: A,L1: list(A),E4: A,L22: 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),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),L22)) )
    <=> ( ( L1 = nil(A) )
        & ( E4 = E3 )
        & ( L22 = nil(A) ) ) ) ).

% list_e_eq_lel(1)
tff(fact_4444_list__e__eq__lel_I2_J,axiom,
    ! [A: $tType,L1: list(A),E4: A,L22: list(A),E3: A] :
      ( ( 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),E4),L22)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) )
    <=> ( ( L1 = nil(A) )
        & ( E4 = E3 )
        & ( L22 = nil(A) ) ) ) ).

% list_e_eq_lel(2)
tff(fact_4445_op__conc__empty__img__id,axiom,
    ! [A: $tType,L6: 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))),L6) = L6 ).

% op_conc_empty_img_id
tff(fact_4446_nth__append__first,axiom,
    ! [A: $tType,I2: nat,L: list(A),L3: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),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),L3)),I2) = aa(nat,A,nth(A,L),I2) ) ) ).

% nth_append_first
tff(fact_4447_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys2)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_4448_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(list(A)),list(list(A)))] :
          ( ( X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
         => ( ! [L4: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A))))
           => ( ! [La: list(A),Acc22: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))
             => ( ! [La: list(A),Acc22: list(list(A)),L4: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A))))
               => ~ ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls))) ) ) ) ) ) ).

% merge_list.cases
tff(fact_4449_list__match__lel__lel,axiom,
    ! [A: $tType,C12: list(A),Qs: A,C23: list(A),C13: list(A),Qs2: A,C24: list(A)] :
      ( ( 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),Qs),C23)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C13),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C24)) )
     => ( ! [C21: list(A)] :
            ( ( C12 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C13),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
           => ( C24 != 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),C23)) ) )
       => ( ( ( C13 = C12 )
           => ( ( Qs2 = Qs )
             => ( C24 != C23 ) ) )
         => ~ ! [C212: list(A)] :
                ( ( C13 = 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),Qs),C212)) )
               => ( C23 != 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),C24)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_4450_foldl__conc__empty__eq,axiom,
    ! [A: $tType,I2: list(A),Ww: list(list(A))] : aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),I2),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),I2),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),nil(A)),Ww)) ).

% foldl_conc_empty_eq
tff(fact_4451_mergesort__by__rel_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ~ ! [R9: fun(A,fun(A,bool)),Xs2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R9),Xs2) ).

% mergesort_by_rel.cases
tff(fact_4452_list__append__eq__Cons__cases,axiom,
    ! [A: $tType,Ys2: list(A),Zs: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
     => ( ( ( Ys2 = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
       => ~ ! [Ys5: list(A)] :
              ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Zs) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_4453_list__Cons__eq__append__cases,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Zs) )
     => ( ( ( Ys2 = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
       => ~ ! [Ys5: list(A)] :
              ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Zs) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_4454_rev__nonempty__induct2_H,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),P: fun(list(A),fun(list(B),bool))] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(B) )
       => ( ! [X3: A,Y3: B] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))
         => ( ! [X3: A,Xs2: list(A),Y3: B] :
                ( ( Xs2 != nil(A) )
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))) )
           => ( ! [X3: A,Y3: B,Ys3: list(B)] :
                  ( ( Ys3 != nil(B) )
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))) )
             => ( ! [X3: A,Xs2: list(A),Y3: B,Ys3: list(B)] :
                    ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
                   => ( ( Xs2 != nil(A) )
                     => ( ( Ys3 != nil(B) )
                       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))) ) ) )
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys2)) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_4455_neq__Nil__rev__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
    <=> ? [Xs3: list(A),X4: A] : L = 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),X4),nil(A))) ) ).

% neq_Nil_rev_conv
tff(fact_4456_rev__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
     => ( ! [X3: A,Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),nil(B)))
       => ( ! [Y3: B,Ys3: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))))
         => ( ! [X3: A,Xs2: list(A),Y3: B,Ys3: list(B)] :
                ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))) )
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys2)) ) ) ) ) ).

% rev_induct2'
tff(fact_4457_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_4458_in__set__list__format,axiom,
    ! [A: $tType,E3: A,L: list(A)] :
      ( pp(aa(set(A),bool,member(A,E3),aa(list(A),set(A),set2(A),L)))
     => ~ ! [L12: list(A),L23: list(A)] : L != 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),E3),L23)) ) ).

% in_set_list_format
tff(fact_4459_xy__in__set__cases,axiom,
    ! [A: $tType,X: A,L: list(A),Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L)))
     => ( pp(aa(set(A),bool,member(A,Y),aa(list(A),set(A),set2(A),L)))
       => ( ( ( X = Y )
           => ! [L12: list(A),L23: list(A)] : L != 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),Y),L23)) )
         => ( ( ( X != Y )
             => ! [L12: list(A),L23: list(A),L32: list(A)] : L != 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),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L32)))) )
           => ~ ( ( X != Y )
               => ! [L12: list(A),L23: list(A),L32: list(A)] : L != 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),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L32)))) ) ) ) ) ) ).

% xy_in_set_cases
tff(fact_4460_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R12: list(A),R23: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4461_set__union__code,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) ).

% set_union_code
tff(fact_4462_distinct__match,axiom,
    ! [A: $tType,Al: list(A),E3: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
      ( 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_4463_lex__append__leftI,axiom,
    ! [A: $tType,Ys2: list(A),Zs: list(A),R3: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys2),Zs)),lex(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3))) ) ).

% lex_append_leftI
tff(fact_4464_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys2)
            & ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
               => ! [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),aa(list(A),set(A),set2(A),Ys2)))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa)) ) ) ) ) ) ).

% sorted_append
tff(fact_4465_list__update__append1,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Ys2: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2),I2,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,I2,X)),Ys2) ) ) ).

% list_update_append1
tff(fact_4466_foldl__rule__aux__P,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2)),P: fun(Sigma2,bool)] :
      ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma_0),L0))
     => ( ! [L12: list(A),L23: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = 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),X3),L23)) )
           => ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L23)))
             => pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),L23)) ) )
       => ( ! [Sigma: Sigma2] :
              ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),nil(A)))
             => pp(aa(Sigma2,bool,P,Sigma)) )
         => pp(aa(Sigma2,bool,P,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0))) ) ) ) ).

% foldl_rule_aux_P
tff(fact_4467_foldl__rule__aux,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2))] :
      ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma_0),L0))
     => ( ! [L12: list(A),L23: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = 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),X3),L23)) )
           => ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L23)))
             => pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),L23)) ) )
       => pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0)),nil(A))) ) ) ).

% foldl_rule_aux
tff(fact_4468_foldl__rule__P,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2)),P: fun(Sigma2,bool)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma_0),nil(A)),L0))
     => ( ! [L12: list(A),L23: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = 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),X3),L23)) )
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L23)))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),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),X3),nil(A)))),L23)) ) )
       => ( ! [Sigma: Sigma2] :
              ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L0),nil(A)))
             => pp(aa(Sigma2,bool,P,Sigma)) )
         => pp(aa(Sigma2,bool,P,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0))) ) ) ) ).

% foldl_rule_P
tff(fact_4469_foldl__rule,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2))] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma_0),nil(A)),L0))
     => ( ! [L12: list(A),L23: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = 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),X3),L23)) )
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L23)))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),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),X3),nil(A)))),L23)) ) )
       => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0)),L0),nil(A))) ) ) ).

% foldl_rule
tff(fact_4470_drop__take__drop__unsplit,axiom,
    ! [A: $tType,I2: nat,J: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,I2,take(A,J,L))),drop(A,J,L)) = drop(A,I2,L) ) ) ).

% drop_take_drop_unsplit
tff(fact_4471_lex__append__left__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys2),Zs)),lex(A,R3))) ) ) ).

% lex_append_left_iff
tff(fact_4472_lex__append__leftD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys2),Zs)),lex(A,R3))) ) ) ).

% lex_append_leftD
tff(fact_4473_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lex(A,R3)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Vs))),lex(A,R3))) ) ) ).

% lex_append_rightI
tff(fact_4474_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),N2: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N2) )
    <=> ? [Ys4: list(A),Y5: A] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N2 ) ) ) ).

% length_Suc_rev_conv
tff(fact_4475_length__compl__rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),L: list(A)] :
      ( pp(aa(list(A),bool,P,nil(A)))
     => ( ! [L4: list(A),E2: A] :
            ( ! [Ll: list(A)] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L4)))
               => pp(aa(list(A),bool,P,Ll)) )
           => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A))))) )
       => pp(aa(list(A),bool,P,L)) ) ) ).

% length_compl_rev_induct
tff(fact_4476_not__distinct__split__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ~ distinct(A,Xs)
     => ~ ! [Y3: A,Ys3: list(A)] :
            ( distinct(A,Ys3)
           => ( pp(aa(set(A),bool,member(A,Y3),aa(list(A),set(A),set2(A),Ys3)))
             => ! [Zs2: list(A)] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))),Zs2)) ) ) ) ).

% not_distinct_split_distinct
tff(fact_4477_filter__eq__snocD,axiom,
    ! [A: $tType,P: fun(A,bool),L: list(A),L3: list(A),X: A] :
      ( ( aa(list(A),list(A),filter2(A,P),L) = 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),X),nil(A))) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L)))
        & pp(aa(A,bool,P,X)) ) ) ).

% filter_eq_snocD
tff(fact_4478_nth__append,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),Ys2: list(A)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),N2) = aa(nat,A,nth(A,Xs),N2) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),N2) = aa(nat,A,nth(A,Ys2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).

% nth_append
tff(fact_4479_list__update__append,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),Ys2: list(A),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2),N2,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,N2,X)),Ys2) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2),N2,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),list_update(A,Ys2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)),X)) ) ) ) ).

% list_update_append
tff(fact_4480_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) )
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
            & ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1)),Ys_2) ) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_4481_slice__prepend,axiom,
    ! [A: $tType,I2: nat,K: nat,Xs: list(A),Ys2: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( slice(A,I2,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(list(A),nat,size_size(list(A)),Ys2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Xs)) ) ) ) ).

% slice_prepend
tff(fact_4482_sorted__append__bigger,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y)) )
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ) ) ) ).

% sorted_append_bigger
tff(fact_4483_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B),Ys2: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Ys2))) ) ).

% horner_sum_append
tff(fact_4484_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),A3),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_4485_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,aa(nat,nat,suc,I2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_4486_take__update__last,axiom,
    ! [A: $tType,N2: nat,List: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),List)))
     => ( list_update(A,take(A,aa(nat,nat,suc,N2),List),N2,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N2,List)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ) ).

% take_update_last
tff(fact_4487_id__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_4488_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),A3: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,Xs,I2,A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_4489_butlast__upd__last__eq,axiom,
    ! [A: $tType,L: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(list(A),nat,size_size(list(A)),L)))
     => ( list_update(A,butlast(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),L)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ) ).

% butlast_upd_last_eq
tff(fact_4490_sort__key__by__quicksort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_sort_key(B,A,F3),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_tf(fun(B,A),fun(list(B),fun(B,bool)),F3),Xs)),Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_tg(fun(B,A),fun(list(B),fun(B,bool)),F3),Xs)),Xs)),aa(list(B),list(B),linorder_sort_key(B,A,F3),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_th(fun(B,A),fun(list(B),fun(B,bool)),F3),Xs)),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_4491_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),aa(list(A),list(A),filter2(A,aTP_Lamp_ti(list(A),fun(A,bool),Xs)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,aTP_Lamp_tj(list(A),fun(A,bool),Xs)),Xs)),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),Xs)))) ) ).

% sort_by_quicksort
tff(fact_4492_drop__last__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(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_4493_Misc_Olast__in__set,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => pp(aa(set(A),bool,member(A,last(A,L)),aa(list(A),set(A),set2(A),L))) ) ).

% Misc.last_in_set
tff(fact_4494_last__drop,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( last(A,drop(A,N2,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_4495_take__butlast__conv,axiom,
    ! [A: $tType,L: 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) = butlast(A,L) ).

% take_butlast_conv
tff(fact_4496_snoc__eq__iff__butlast_H,axiom,
    ! [A: $tType,Ys2: list(A),Xs: list(A),X: A] :
      ( ( Ys2 = 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))) )
    <=> ( ( Ys2 != nil(A) )
        & ( butlast(A,Ys2) = Xs )
        & ( last(A,Ys2) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_4497_distinct__butlast__swap,axiom,
    ! [A: $tType,Pq: list(A),I2: nat] :
      ( distinct(A,Pq)
     => distinct(A,butlast(A,list_update(A,Pq,I2,last(A,Pq)))) ) ).

% distinct_butlast_swap
tff(fact_4498_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,aTP_Lamp_tk(B,fun(A,B),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_4499_butlast__update_H,axiom,
    ! [A: $tType,L: list(A),I2: nat,X: A] : list_update(A,butlast(A,L),I2,X) = butlast(A,list_update(A,L,I2,X)) ).

% butlast_update'
tff(fact_4500_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),filter2(A,aa(B,fun(A,bool),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,bool)),F3),K)),aa(list(A),list(A),linorder_sort_key(A,B,F3),Xs)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,bool),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,bool)),F3),K)),Xs) ) ).

% sort_key_stable
tff(fact_4501_last__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ( Xs != nil(A) )
     => ( pp(aa(A,bool,P,last(A,Xs)))
       => ( last(A,aa(list(A),list(A),filter2(A,P),Xs)) = last(A,Xs) ) ) ) ).

% last_filter
tff(fact_4502_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),Xs)) ) ).

% sorted_sort
tff(fact_4503_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_4504_butlast__subset,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] :
      ( ( Xs != nil(A) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),butlast(A,Xs))),A5)) ) ) ).

% butlast_subset
tff(fact_4505_butlast__eq__cons__conv,axiom,
    ! [A: $tType,L: list(A),X: A,Xs: list(A)] :
      ( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
    <=> ? [Xl: A] : L = 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)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl),nil(A)))) ) ).

% butlast_eq_cons_conv
tff(fact_4506_butlast__eq__consE,axiom,
    ! [A: $tType,L: list(A),X: A,Xs: list(A)] :
      ( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
     => ~ ! [Xl2: A] : L != 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)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl2),nil(A)))) ) ).

% butlast_eq_consE
tff(fact_4507_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),butlast(A,Xs)) ) ) ) ).

% sorted_butlast
tff(fact_4508_nth__butlast,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),butlast(A,Xs))))
     => ( aa(nat,A,nth(A,butlast(A,Xs)),N2) = aa(nat,A,nth(A,Xs),N2) ) ) ).

% nth_butlast
tff(fact_4509_take__butlast,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,N2,butlast(A,Xs)) = take(A,N2,Xs) ) ) ).

% take_butlast
tff(fact_4510_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),aa(list(A),list(A),remdups(A),Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_4511_remdup__sort__mergesort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),remdups(A)),linorder_sort_key(A,A,aTP_Lamp_rm(A,A))) = mergesort_remdups(A) ) ) ).

% remdup_sort_mergesort_remdups
tff(fact_4512_butlast__take,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( butlast(A,take(A,N2,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_4513_take__minus__one__conv__butlast,axiom,
    ! [A: $tType,N2: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),L)))
     => ( take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,take(A,N2,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_4514_last__take__nth__conv,axiom,
    ! [A: $tType,N2: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),L)))
     => ( ( N2 != zero_zero(nat) )
       => ( last(A,take(A,N2,L)) = aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) ) ) ) ).

% last_take_nth_conv
tff(fact_4515_sort__key__by__quicksort__code,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = aa(list(B),list(B),case_list(list(B),B,nil(B),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_tq(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),F3),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_4516_upto__aux__rec,axiom,
    ! [J: int,I2: int,Js: list(int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( upto_aux(I2,J,Js) = Js ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( upto_aux(I2,J,Js) = upto_aux(I2,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_4517_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_tr(A,fun(A,bool),X)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),ord_less_eq(A),X)),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_4518_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
         => ( ( ( X = nil(A) )
             => ( Y != nil(A) ) )
           => ~ ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( Y != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_tr(A,fun(A,bool),X3)),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),ord_less_eq(A),X3)),Xs2)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_4519_list_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] : aa(B,C,H,aa(list(A),B,case_list(B,A,F1,F22),List)) = aa(list(A),C,case_list(C,A,aa(B,C,H,F1),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_ts(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H),F22)),List) ).

% list.case_distrib
tff(fact_4520_sort__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_rm(A,A)) = linorder_quicksort(A) ) ) ).

% sort_quicksort
tff(fact_4521_sorted__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),linorder_quicksort(A),Xs)) ) ).

% sorted_quicksort
tff(fact_4522_remdups__adj__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),aTP_Lamp_tt(A,fun(A,fun(list(A),list(A))),X)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_4523_part__code_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Pivot: A] : linorder_part(B,A,F3,Pivot,nil(B)) = aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),nil(B)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),nil(B))) ) ).

% part_code(1)
tff(fact_4524_part__code_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Pivot: A,X: B,Xs: list(B)] : linorder_part(B,A,F3,Pivot,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(product_prod(list(B),product_prod(list(B),list(B))),product_prod(list(B),product_prod(list(B),list(B))),aa(fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),fun(product_prod(list(B),product_prod(list(B),list(B))),product_prod(list(B),product_prod(list(B),list(B)))),product_case_prod(list(B),product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_tv(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),F3),Pivot),X)),linorder_part(B,A,F3,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_4525_part__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Pivot: A,Xs: list(B)] : linorder_part(B,A,F3,Pivot,Xs) = aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),aa(list(B),list(B),filter2(B,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,A),fun(A,fun(B,bool)),F3),Pivot)),Xs)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),aa(list(B),list(B),filter2(B,aa(A,fun(B,bool),aTP_Lamp_tx(fun(B,A),fun(A,fun(B,bool)),F3),Pivot)),Xs)),aa(list(B),list(B),filter2(B,aa(A,fun(B,bool),aTP_Lamp_ty(fun(B,A),fun(A,fun(B,bool)),F3),Pivot)),Xs))) ) ).

% part_def
tff(fact_4526_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
         => ( pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),X))
           => ( ( ( X = nil(A) )
               => ( ( Y = nil(A) )
                 => ~ pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),nil(A))) ) )
             => ~ ! [X3: A,Xs2: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aTP_Lamp_tr(A,fun(A,bool),X3)),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),ord_less_eq(A),X3)),Xs2)))) )
                     => ~ pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_4527_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J)))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
         => ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
         => ( upto(I2,J) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_4528_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)))
       => ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
               => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
               => ( Y = nil(int) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))) ) ) ) ).

% upto.pelims
tff(fact_4529_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P) = aa(list(A),A,case_list(A,A,abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P),aTP_Lamp_tz(A,fun(list(A),A))),aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_4530_sort__upto,axiom,
    ! [I2: int,J: int] : aa(list(int),list(int),linorder_sort_key(int,int,aTP_Lamp_gs(int,int)),upto(I2,J)) = upto(I2,J) ).

% sort_upto
tff(fact_4531_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil(int) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2)) ) ).

% upto_Nil
tff(fact_4532_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil(int) = upto(I2,J) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2)) ) ).

% upto_Nil2
tff(fact_4533_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
     => ( upto(I2,J) = nil(int) ) ) ).

% upto_empty
tff(fact_4534_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K))),J))
     => ( aa(nat,int,nth(int,upto(I2,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_4535_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N2)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N2)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N2))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N2)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N2)) = nil(int) ) ) ) ).

% upto_rec_numeral(1)
tff(fact_4536_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))) = nil(int) ) ) ) ).

% upto_rec_numeral(4)
tff(fact_4537_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N2)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N2)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N2))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N2)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N2)) = nil(int) ) ) ) ).

% upto_rec_numeral(3)
tff(fact_4538_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N2))) = nil(int) ) ) ) ).

% upto_rec_numeral(2)
tff(fact_4539_sorted__upto,axiom,
    ! [M: int,N2: int] : sorted_wrt(int,ord_less_eq(int),upto(M,N2)) ).

% sorted_upto
tff(fact_4540_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : sorted_wrt(int,ord_less(int),upto(I2,J)) ).

% sorted_wrt_upto
tff(fact_4541_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fFalse,aTP_Lamp_ua(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(2)
tff(fact_4542_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fTrue,aTP_Lamp_ub(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(1)
tff(fact_4543_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_4544_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_4545_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
         => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
         => ( Y = nil(int) ) ) ) ) ).

% upto.elims
tff(fact_4546_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
       => ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
       => ( upto(I2,J) = nil(int) ) ) ) ).

% upto.simps
tff(fact_4547_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_4548_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( upto(I2,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,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_4549_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_4550_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys2: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys2),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & pp(aa(A,bool,P,Y))
        & ~ ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Ys2)))
              & pp(aa(A,bool,P,X4)) ) ) ) ).

% extract_Some_iff
tff(fact_4551_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys2: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys2),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & pp(aa(A,bool,P,Y))
        & ~ ? [X5: A] :
              ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Ys2)))
              & pp(aa(A,bool,P,X5)) ) ) ) ).

% extract_SomeE
tff(fact_4552_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
      ( pp(aa(product_prod(list(A),list(A)),bool,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)))
     => ( ! [Ys3: list(A)] :
            ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
       => ( ! [X3: A,Xs2: list(A),Ys3: list(A)] :
              ( pp(aa(product_prod(list(A),list(A)),bool,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),X3),Xs2)),Ys3)))
             => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Ys3),Xs2))
               => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3)) ) )
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ).

% splice.pinduct
tff(fact_4553_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),X),Xs))) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_ud(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X)),extract(A,P,Xs)) ) ) ) ).

% extract_Cons_code
tff(fact_4554_subset__eq__mset__impl_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: option(bool)] :
      ( ( subset_eq_mset_impl(A,X,Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa2),nil(A)))) ) )
       => ~ ! [X3: A,Xs2: list(A)] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ( Y != case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_uf(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2)),extract(A,aa(A,fun(A,bool),fequal(A),X3),Xa2)) ) ) ) ) ).

% subset_eq_mset_impl.elims
tff(fact_4555_splice_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: list(A)] :
      ( ( splice(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa2)))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa2 )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,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)),Xa2))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),splice(A,Xa2,Xs2)) )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,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),X3),Xs2)),Xa2))) ) ) ) ) ) ).

% splice.pelims
tff(fact_4556_Id__on__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( id_on(A,A5) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_ug(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A5) ) ) ).

% Id_on_fold
tff(fact_4557_Id__on__def,axiom,
    ! [A: $tType,A5: set(A)] : id_on(A,A5) = 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_uh(A,set(product_prod(A,A)))),A5)) ).

% Id_on_def
tff(fact_4558_Id__onI,axiom,
    ! [A: $tType,A3: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id_on(A,A5))) ) ).

% Id_onI
tff(fact_4559_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_4560_Id__on__iff,axiom,
    ! [A: $tType,X: A,Y: A,A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),id_on(A,A5)))
    <=> ( ( X = Y )
        & pp(aa(set(A),bool,member(A,X),A5)) ) ) ).

% Id_on_iff
tff(fact_4561_Id__on__eqI,axiom,
    ! [A: $tType,A3: A,B2: A,A5: set(A)] :
      ( ( A3 = B2 )
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id_on(A,A5))) ) ) ).

% Id_on_eqI
tff(fact_4562_Id__onE,axiom,
    ! [A: $tType,C2: product_prod(A,A),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),C2),id_on(A,A5)))
     => ~ ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ) ).

% Id_onE
tff(fact_4563_subset__eq__mset__impl_Osimps_I1_J,axiom,
    ! [A: $tType,Ys2: list(A)] : subset_eq_mset_impl(A,nil(A),Ys2) = aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Ys2),nil(A)))) ).

% subset_eq_mset_impl.simps(1)
tff(fact_4564_Id__on__def_H,axiom,
    ! [A: $tType,A5: fun(A,bool)] : id_on(A,aa(fun(A,bool),set(A),collect(A),A5)) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ui(fun(A,bool),fun(A,fun(A,bool)),A5))) ).

% Id_on_def'
tff(fact_4565_subset__eq__mset__impl_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A)] : subset_eq_mset_impl(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2) = case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_uf(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs)),extract(A,aa(A,fun(A,bool),fequal(A),X),Ys2)) ).

% subset_eq_mset_impl.simps(2)
tff(fact_4566_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,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)),Ys2)))
     => ( splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),splice(A,Ys2,Xs)) ) ) ).

% splice.psimps(2)
tff(fact_4567_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys2: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)))
     => ( splice(A,nil(A),Ys2) = Ys2 ) ) ).

% splice.psimps(1)
tff(fact_4568_subset__eq__mset__impl_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: option(bool)] :
      ( ( subset_eq_mset_impl(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa2)))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa2),nil(A)))) )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,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)),Xa2))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( ( Y = case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_uf(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2)),extract(A,aa(A,fun(A,bool),fequal(A),X3),Xa2)) )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,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),X3),Xs2)),Xa2))) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
tff(fact_4569_merge__list__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ls2: list(list(A)),As: list(list(A))] :
          ( ! [L4: list(A)] :
              ( pp(aa(set(list(A)),bool,member(list(A),L4),aa(list(list(A)),set(list(A)),set2(list(A)),Ls2)))
             => ( distinct(A,L4)
                & sorted_wrt(A,ord_less_eq(A),L4) ) )
         => ( ! [L4: list(A)] :
                ( pp(aa(set(list(A)),bool,member(list(A),L4),aa(list(list(A)),set(list(A)),set2(list(A)),As)))
               => ( distinct(A,L4)
                  & sorted_wrt(A,ord_less_eq(A),L4) ) )
           => ( distinct(A,merge_list(A,As,Ls2))
              & sorted_wrt(A,ord_less_eq(A),merge_list(A,As,Ls2))
              & ( aa(list(A),set(A),set2(A),merge_list(A,As,Ls2)) = aa(list(A),set(A),set2(A),concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),As),Ls2))) ) ) ) ) ) ).

% merge_list_correct
tff(fact_4570_take__hd__drop,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),drop(A,N2,Xs))),nil(A))) = take(A,aa(nat,nat,suc,N2),Xs) ) ) ).

% take_hd_drop
tff(fact_4571_Succ__def,axiom,
    ! [A: $tType,Kl: set(list(A)),Kl2: list(A)] : bNF_Greatest_Succ(A,Kl,Kl2) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_uj(set(list(A)),fun(list(A),fun(A,bool)),Kl),Kl2)) ).

% Succ_def
tff(fact_4572_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J))
     => ( aa(list(A),A,hd(A),take(A,J,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_take
tff(fact_4573_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_4574_merge__list_Osimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: 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),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) ) ).

% merge_list.simps(4)
tff(fact_4575_merge__list_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: 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),Acc23),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) ) ).

% merge_list.simps(3)
tff(fact_4576_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_4577_merge__list_Osimps_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Acc23: list(list(A)),L1: list(A),L22: list(A),Ls2: list(list(A))] : merge_list(A,Acc23,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),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),Acc23),Ls2) ) ).

% merge_list.simps(5)
tff(fact_4578_sorted__hd__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ! [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,hd(A),Xs)),X5)) ) ) ) ) ).

% sorted_hd_min
tff(fact_4579_hd__drop__conv__nth,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),A,hd(A),drop(A,N2,Xs)) = aa(nat,A,nth(A,Xs),N2) ) ) ).

% hd_drop_conv_nth
tff(fact_4580_sorted__hd__last,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( ( L != nil(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,hd(A),L)),last(A,L))) ) ) ) ).

% sorted_hd_last
tff(fact_4581_hd__last__singletonI,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( aa(list(A),A,hd(A),Xs) = last(A,Xs) )
       => ( distinct(A,Xs)
         => ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),nil(A)) ) ) ) ) ).

% hd_last_singletonI
tff(fact_4582_hd__butlast,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),A,hd(A),butlast(A,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_butlast
tff(fact_4583_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),A,hd(A),slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).

% slice_head
tff(fact_4584_merge__list_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(list(A)),Xa2: list(list(A)),Y: list(A)] :
          ( ( merge_list(A,X,Xa2) = Y )
         => ( ( ( X = nil(list(A)) )
             => ( ( Xa2 = nil(list(A)) )
               => ( Y != nil(A) ) ) )
           => ( ( ( X = nil(list(A)) )
               => ! [L4: list(A)] :
                    ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A))) )
                   => ( Y != L4 ) ) )
             => ( ! [La: list(A),Acc22: list(list(A))] :
                    ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                   => ( ( Xa2 = nil(list(A)) )
                     => ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)) ) ) )
               => ( ! [La: list(A),Acc22: list(list(A))] :
                      ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                     => ! [L4: list(A)] :
                          ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),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)),L4),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))) ) ) )
                 => ~ ! [L12: list(A),L23: list(A),Ls: list(list(A))] :
                        ( ( Xa2 = 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),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,L12,L23)),X),Ls) ) ) ) ) ) ) ) ) ).

% merge_list.elims
tff(fact_4585_merge__list_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(list(A)),Xa2: list(list(A)),Y: list(A)] :
          ( ( merge_list(A,X,Xa2) = Y )
         => ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),Xa2)))
           => ( ( ( X = nil(list(A)) )
               => ( ( Xa2 = nil(list(A)) )
                 => ( ( Y = nil(A) )
                   => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))) ) ) )
             => ( ( ( X = nil(list(A)) )
                 => ! [L4: list(A)] :
                      ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A))) )
                     => ( ( Y = L4 )
                       => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,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)),L4),nil(list(A)))))) ) ) )
               => ( ! [La: list(A),Acc22: list(list(A))] :
                      ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                     => ( ( Xa2 = nil(list(A)) )
                       => ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)) )
                         => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A))))) ) ) )
                 => ( ! [La: list(A),Acc22: list(list(A))] :
                        ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                       => ! [L4: list(A)] :
                            ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),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)),L4),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))) )
                             => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A)))))) ) ) )
                   => ~ ! [L12: list(A),L23: list(A),Ls: list(list(A))] :
                          ( ( Xa2 = 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),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,L12,L23)),X),Ls) )
                           => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls))))) ) ) ) ) ) ) ) ) ) ).

% merge_list.pelims
tff(fact_4586_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),A5: set(A),B4: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),S))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),S))
         => ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = finite_fold(A,B,F3,finite_fold(A,B,F3,Z2,A5),B4) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_4587_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),A5: set(A),X: A,Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( finite_fold(A,B,F3,Z2,A5) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_4588_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => finite4664212375090638736ute_on(A,B,S,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_uk(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_4589_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(A,fun(B,B)),A5: set(A),S2: B,T6: B,B4: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( finite4664212375090638736ute_on(A,B,S,G3)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),S))
         => ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A5))
                 => ( aa(A,fun(B,B),F3,X3) = aa(A,fun(B,B),G3,X3) ) )
             => ( ( S2 = T6 )
               => ( ( A5 = B4 )
                 => ( finite_fold(A,B,F3,S2,A5) = finite_fold(A,B,G3,T6,B4) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_4590_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A5: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z2,A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z2),A5) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_4591_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A5: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ( finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z2),A5) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_4592_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A5: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ( finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z2,A5)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_4593_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R2: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G3),top_top(set(C)))),S))
       => finite4664212375090638736ute_on(C,B,R2,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_4594_merge__list_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,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_4595_merge__list_Opsimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,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_4596_merge__list_Opsimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A))] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),nil(list(A)))))
         => ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) ) ) ) ).

% merge_list.psimps(3)
tff(fact_4597_merge__list_Opsimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A)),L: list(A)] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A))))))
         => ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) ) ) ) ).

% merge_list.psimps(4)
tff(fact_4598_merge__list_Opinduct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A0: list(list(A)),A1: list(list(A)),P: fun(list(list(A)),fun(list(list(A)),bool))] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,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)))
         => ( ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,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)))))
             => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),nil(list(A)))) )
           => ( ! [L4: list(A)] :
                  ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,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)),L4),nil(list(A))))))
                 => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A))))) )
             => ( ! [La: list(A),Acc22: list(list(A))] :
                    ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))))
                   => ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)))
                     => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))) ) )
               => ( ! [La: list(A),Acc22: list(list(A)),L4: list(A)] :
                      ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A))))))
                     => ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))))
                       => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L4),nil(list(A))))) ) )
                 => ( ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls: list(list(A))] :
                        ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,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),Ls)))))
                       => ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,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)),Ls))
                         => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,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),Ls)))) ) )
                   => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,A0),A1)) ) ) ) ) ) ) ) ).

% merge_list.pinduct
tff(fact_4599_merge__list_Opsimps_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Acc23: list(list(A)),L1: list(A),L22: list(A),Ls2: list(list(A))] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2)))))
         => ( merge_list(A,Acc23,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),Acc23),Ls2) ) ) ) ).

% merge_list.psimps(5)
tff(fact_4600_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A5: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_4601_insort__key__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [A3: B,Xs: list(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,A3),aa(list(B),set(B),set2(B),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
           => ( ( aa(list(B),B,hd(B),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ul(B,fun(fun(B,A),fun(B,bool)),A3),F3)),Xs)) = A3 )
             => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),A3),remove1(B,A3,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_4602_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),N2: A,Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N2),Ns))),lenlex(A,R3)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N2)),R3)) )
        | ( ( M = N2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R3))) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4603_sort__mergesort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_rm(A,A)) = mergesort(A) ) ) ).

% sort_mergesort
tff(fact_4604_map__ident,axiom,
    ! [A: $tType,X5: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_cq(A,A)),X5) = X5 ).

% map_ident
tff(fact_4605_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ns)),lenlex(A,R3)))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_4606_nth__map,axiom,
    ! [B: $tType,A: $tType,N2: nat,Xs: list(A),F3: fun(A,B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F3),Xs)),N2) = aa(A,B,F3,aa(nat,A,nth(A,Xs),N2)) ) ) ).

% nth_map
tff(fact_4607_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_um(fun(B,A),fun(B,list(A)),F3)),Xs)) = aa(list(B),list(A),map(B,A,F3),Xs) ).

% concat_map_singleton
tff(fact_4608_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G3: fun(A,fun(B,A)),A3: A,F3: fun(C,B),Xs: list(C)] : aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,G3),A3),aa(list(C),list(B),map(C,B,F3),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_un(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G3),F3)),A3),Xs) ).

% foldl_map
tff(fact_4609_list_Omap__ident,axiom,
    ! [A: $tType,T6: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_cq(A,A)),T6) = T6 ).

% list.map_ident
tff(fact_4610_distinct__mapI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => distinct(B,L) ) ).

% distinct_mapI
tff(fact_4611_map__consI_I1_J,axiom,
    ! [A: $tType,B: $tType,W2: list(A),F3: fun(B,A),Ww: list(B),A3: B] :
      ( ( W2 = aa(list(B),list(A),map(B,A,F3),Ww) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F3,A3)),W2) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww)) ) ) ).

% map_consI(1)
tff(fact_4612_map__eq__consE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls2: list(B),Fa: A,Fl: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
     => ~ ! [A6: B,L4: list(B)] :
            ( ( Ls2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A6),L4) )
           => ( ( aa(B,A,F3,A6) = Fa )
             => ( aa(list(B),list(A),map(B,A,F3),L4) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_4613_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B),L3: list(B),I2: nat] :
      ( ( aa(list(B),list(A),map(B,A,F3),L) = aa(list(B),list(A),map(B,A,F3),L3) )
     => ( aa(B,A,F3,aa(nat,B,nth(B,L),I2)) = aa(B,A,F3,aa(nat,B,nth(B,L3),I2)) ) ) ).

% map_eq_nth_eq
tff(fact_4614_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,bool)),F3: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R2,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_uo(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R2),F3),Xs) ) ).

% sorted_wrt_map
tff(fact_4615_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : product(A,B,Xs,Ys2) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_up(list(B),fun(A,list(product_prod(A,B))),Ys2)),Xs)) ).

% product_concat_map
tff(fact_4616_append__eq__mapE,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F3: fun(B,A),Ls2: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F3),Ls2) )
     => ~ ! [L4: list(B),L7: list(B)] :
            ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L7) )
           => ( ( aa(list(B),list(A),map(B,A,F3),L4) = Fl )
             => ( aa(list(B),list(A),map(B,A,F3),L7) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_4617_map__eq__appendE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls2: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
     => ~ ! [L4: list(B),L7: list(B)] :
            ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L7) )
           => ( ( aa(list(B),list(A),map(B,A,F3),L4) = Fl )
             => ( aa(list(B),list(A),map(B,A,F3),L7) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_4618_Misc_Oappend__eq__map__conv,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F3: fun(B,A),Ls2: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F3),Ls2) )
    <=> ? [L2: list(B),L8: list(B)] :
          ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L8) )
          & ( aa(list(B),list(A),map(B,A,F3),L2) = Fl )
          & ( aa(list(B),list(A),map(B,A,F3),L8) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_4619_Misc_Omap__eq__append__conv,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls2: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
    <=> ? [L2: list(B),L8: list(B)] :
          ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L8) )
          & ( aa(list(B),list(A),map(B,A,F3),L2) = Fl )
          & ( aa(list(B),list(A),map(B,A,F3),L8) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_4620_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N2),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_ur(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N2,Xs))) ).

% n_lists.simps(2)
tff(fact_4621_set__oo__map__alt,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X5: list(A)] : aa(list(A),set(B),aa(fun(list(A),list(B)),fun(list(A),set(B)),comp(list(B),set(B),list(A),set2(B)),map(A,B,F3)),X5) = aa(set(A),set(B),image2(A,B,F3),aa(list(A),set(A),set2(A),X5)) ).

% set_oo_map_alt
tff(fact_4622_map__consI_I2_J,axiom,
    ! [B: $tType,A: $tType,W2: list(A),L: list(A),F3: 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),map(B,A,F3),Ww)),L) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F3,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),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww))),L) ) ) ).

% map_consI(2)
tff(fact_4623_distinct__map__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B),X: B,Y: B] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => ( ( aa(B,A,F3,X) = aa(B,A,F3,Y) )
       => ( pp(aa(set(B),bool,member(B,X),aa(list(B),set(B),set2(B),L)))
         => ( pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),L)))
           => ( X = Y ) ) ) ) ) ).

% distinct_map_eq
tff(fact_4624_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
        <=> sorted_wrt(B,aTP_Lamp_us(fun(B,A),fun(B,fun(B,bool)),F3),Xs) ) ) ).

% sorted_map
tff(fact_4625_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P: fun(B,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs))) ) ) ).

% sorted_filter
tff(fact_4626_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ).

% sorted_insort_key
tff(fact_4627_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs))) ) ).

% sorted_sort_key
tff(fact_4628_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),remove1(B,X,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_4629_lenlex__irreflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lenlex(A,R3))) ) ).

% lenlex_irreflexive
tff(fact_4630_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ns),nil(A))),lenlex(A,R3))) ).

% Nil_lenlex_iff2
tff(fact_4631_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys2: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),Ys2)),product(A,B,Xs,Ys2)) ).

% product.simps(2)
tff(fact_4632_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_ut(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_si(list(A),bool)),Xss)) ).

% transpose_aux_filter_head
tff(fact_4633_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),G3: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_uu(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F3),G3),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_4634_Id__on__set,axiom,
    ! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_uv(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_4635_distinct__idx,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),L: list(B),I2: nat,J: nat] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),L)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L)))
         => ( ( aa(B,A,F3,aa(nat,B,nth(B,L),I2)) = aa(B,A,F3,aa(nat,B,nth(B,L),J)) )
           => ( I2 = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4636_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I2: nat,L: list(A),F3: fun(A,B),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
       => ( aa(A,B,F3,aa(nat,A,nth(A,L),I2)) = aa(A,B,F3,X) ) )
     => ( aa(list(A),list(B),map(A,B,F3),list_update(A,L,I2,X)) = aa(list(A),list(B),map(A,B,F3),L) ) ) ).

% map_upd_eq
tff(fact_4637_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P: fun(B,bool),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => ( pp(aa(B,bool,P,X))
           => ( aa(list(B),list(B),filter2(B,P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),filter2(B,P),Xs)) ) ) ) ) ).

% filter_insort
tff(fact_4638_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : subseqs(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),subseqs(A,Xs))),subseqs(A,Xs)) ).

% subseqs.simps(2)
tff(fact_4639_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R3)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).

% lenlex_length
tff(fact_4640_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R2: set(product_prod(A,A)),Vs: list(A),Ys2: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs)),lenlex(A,R2)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys2) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2))),lenlex(A,R2))) ) ) ).

% lenlex_append1
tff(fact_4641_map__by__foldl,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),L: list(A)] : aa(list(A),list(B),aa(list(B),fun(list(A),list(B)),foldl(list(B),A,aTP_Lamp_uw(fun(A,B),fun(list(B),fun(A,list(B))),F3)),nil(B)),L) = aa(list(A),list(B),map(A,B,F3),L) ).

% map_by_foldl
tff(fact_4642_mergesort__remdups__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),mergesort_remdups(A),Xs) = merge_list(A,nil(list(A)),aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_ux(A,list(A))),Xs)) ) ).

% mergesort_remdups_def
tff(fact_4643_map__distinct__upd__conv,axiom,
    ! [B: $tType,A: $tType,I2: nat,L: list(A),F3: fun(A,B),X: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( distinct(A,L)
       => ( list_update(B,aa(list(A),list(B),map(A,B,F3),L),I2,X) = aa(list(A),list(B),map(A,B,fun_upd(A,B,F3,aa(nat,A,nth(A,L),I2),X)),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4644_lenlex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_uy(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lenlex_conv
tff(fact_4645_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ~ ! [L4: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L4))
               => ( ( aa(list(B),set(B),set2(B),L4) = A5 )
                 => ( aa(list(B),nat,size_size(list(B)),L4) != aa(set(B),nat,finite_card(B),A5) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_4646_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F3: fun(nat,B),Ns: list(nat)] :
          ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F3,X3)),aa(nat,B,F3,Y3))) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(B,aa(list(nat),list(B),map(nat,B,F3),Ns)))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4647_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_uz(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F3),P),Xs) ).

% map_filter_map_filter
tff(fact_4648_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_va(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_4649_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),groups8242544230860333062m_list(A,Xs))) ) ) ).

% member_le_sum_list
tff(fact_4650_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),aa(list(A),list(A),filter2(A,P),Xs)))),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)))) ).

% sum_list_filter_le_nat
tff(fact_4651_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_du(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G3),Xs))) ) ).

% sum_list_addf
tff(fact_4652_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),C2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_dx(fun(B,A),fun(A,fun(B,A)),F3),C2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),C2) ) ).

% sum_list_mult_const
tff(fact_4653_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F3: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dy(A,fun(fun(B,A),fun(B,A)),C2),F3)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))) ) ).

% sum_list_const_mult
tff(fact_4654_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dz(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G3),Xs))) ) ).

% sum_list_subtractf
tff(fact_4655_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_option(list(A),A,map_filter(B,A,F3,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_vb(fun(B,option(A)),fun(list(B),fun(A,list(A))),F3),Xs),aa(B,option(A),F3,X)) ).

% map_filter_simps(1)
tff(fact_4656_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xs))) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_4657_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => ( ( groups8242544230860333062m_list(A,Xs) = zero_zero(A) )
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_4658_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups8242544230860333062m_list(A,Xs)),zero_zero(A))) ) ) ).

% sum_list_nonpos
tff(fact_4659_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs))),groups8242544230860333062m_list(A,aa(list(A),list(A),map(A,A,abs_abs(A)),Xs)))) ) ).

% sum_list_abs
tff(fact_4660_sum__list__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_ld(fun(A,nat),fun(A,nat),F3)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_4661_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))) )
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs)))) ) ) ).

% sum_list_mono
tff(fact_4662_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F3: fun(B,A),P: fun(B,bool),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_vc(fun(B,A),fun(fun(B,bool),fun(B,A)),F3),P)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_4663_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(A,Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_vd(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_4664_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs)))) ) ) ) ).

% sum_list_strict_mono
tff(fact_4665_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),groups8242544230860333062m_list(A,Ns))) ) ) ).

% elem_le_sum_list
tff(fact_4666_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R3: B,Xs: list(C)] : groups8242544230860333062m_list(B,aa(list(C),list(B),map(C,B,aTP_Lamp_ve(B,fun(C,B),R3)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R3) ) ).

% sum_list_triv
tff(fact_4667_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_rm(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_4668_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_vf(fun(A,nat),fun(list(A),fun(A,nat)),F3),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_4669_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : groups8242544230860333062m_list(B,Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_4670_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_vg(nat,fun(nat,fun(list(nat),bool)),M),N7))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_vh(nat,fun(nat,fun(list(nat),bool)),M),N7)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_vi(nat,fun(nat,fun(list(nat),bool)),M),N7)))) ) ) ).

% card_length_sum_list_rec
tff(fact_4671_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_vg(nat,fun(nat,fun(list(nat),bool)),M),N7))) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N7),M)),one_one(nat))),N7) ).

% card_length_sum_list
tff(fact_4672_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs,K,X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),X)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_4673_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_vj(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F3)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_4674_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys2)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_up(list(B),fun(A,list(product_prod(A,B))),Ys2)),Xs))) ).

% product_code
tff(fact_4675_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),bool)] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),A0))
     => ( ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A))))
         => pp(aa(list(list(A)),bool,P,nil(list(A)))) )
       => ( ! [Xss2: list(list(A))] :
              ( pp(aa(list(list(A)),bool,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)))
             => ( pp(aa(list(list(A)),bool,P,Xss2))
               => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
         => ( ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2)))
               => ( pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_vk(A,fun(list(A),list(list(A)))))),Xss2)))))
                 => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) )
           => pp(aa(list(list(A)),bool,P,A0)) ) ) ) ) ).

% transpose.pinduct
tff(fact_4676_transpose_Oelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( ( ( X = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_ut(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_vk(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_4677_nth__transpose,axiom,
    ! [A: $tType,I2: nat,Xs: list(list(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_vl(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_vm(nat,fun(list(A),bool),I2)),Xs)) ) ) ).

% nth_transpose
tff(fact_4678_transpose_Opelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),X))
       => ( ( ( X = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A)))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
           => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_ut(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_vk(A,fun(list(A),list(list(A)))))),Xss2))))) )
                   => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_4679_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_ut(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_vk(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_4680_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_ut(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_vk(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_4681_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),Xss)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_vn(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_4682_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L))
              & ( aa(list(B),set(B),set2(B),L) = A5 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A5) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A5) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_4683_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_vk(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_si(list(A),bool)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_4684_in__hd__or__tl__conv,axiom,
    ! [A: $tType,L: list(A),X: A] :
      ( ( L != nil(A) )
     => ( ( ( X = aa(list(A),A,hd(A),L) )
          | pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),L)))) )
      <=> pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L))) ) ) ).

% in_hd_or_tl_conv
tff(fact_4685_in__set__tlD,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))))
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_tlD
tff(fact_4686_tl__obtain__elem,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
       => ~ ! [E2: A] : Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) ) ) ).

% tl_obtain_elem
tff(fact_4687_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),tl(A),Xs)) ) ) ).

% sorted_tl
tff(fact_4688_not__hd__in__tl,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( X != aa(list(A),A,hd(A),Xs) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs)))) ) ) ).

% not_hd_in_tl
tff(fact_4689_tl__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) != nil(A) )
     => ( last(A,Xs) = last(A,aa(list(A),list(A),tl(A),Xs)) ) ) ).

% tl_last
tff(fact_4690_tl__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_vo(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_4691_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys2),aTP_Lamp_vp(list(A),fun(A,fun(list(A),list(A))),Ys2)),Xs) ).

% tl_append
tff(fact_4692_tl__subset,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] :
      ( ( Xs != nil(A) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))),A5)) ) ) ).

% tl_subset
tff(fact_4693_Misc_Onth__tl,axiom,
    ! [A: $tType,Xs: list(A),N2: nat] :
      ( ( Xs != nil(A) )
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N2) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N2)) ) ) ).

% Misc.nth_tl
tff(fact_4694_list__take__induct__tl2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),P: fun(B,fun(A,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(B,fun(A,bool),P,aa(nat,B,nth(B,Ys2),N5)),aa(nat,A,nth(A,Xs),N5))) )
       => ! [N8: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N8),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
           => pp(aa(A,bool,aa(B,fun(A,bool),P,aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Ys2)),N8)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N8))) ) ) ) ).

% list_take_induct_tl2
tff(fact_4695_distinct__hd__tl,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( ( X = aa(list(A),A,hd(A),Xs) )
       => ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs)))) ) ) ).

% distinct_hd_tl
tff(fact_4696_remove1__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( remove1(A,aa(list(A),A,hd(A),Xs),Xs) = aa(list(A),list(A),tl(A),Xs) ) ) ).

% remove1_tl
tff(fact_4697_List_Onth__tl,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N2) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N2)) ) ) ).

% List.nth_tl
tff(fact_4698_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_4699_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B),B4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),S))
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A5) = sorted8670434370408473282of_set(A,B,Less_eq,F3,B4) )
           => ( pp(aa(set(B),bool,finite_finite2(B),A5))
             => ( pp(aa(set(B),bool,finite_finite2(B),B4))
               => ( A5 = B4 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_4700_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( aa(list(B),set(B),set2(B),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) = A5 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_4701_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => ( aa(list(B),nat,size_size(list(B)),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) = aa(set(B),nat,finite_card(B),A5) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_4702_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => distinct(A,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_4703_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_4704_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_4705_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A5) = nil(B) )
          <=> ( A5 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4706_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S))
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),Xs))
         => ( distinct(B,Xs)
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_4707_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),X: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B))))) = remove1(B,X,sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_4708_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),X: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)) = insort_key(A,B,Less_eq,F3,X,sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_4709_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_vm(nat,fun(list(A),bool),I2)),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I2) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_4710_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_vl(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_vm(nat,fun(list(A),bool),I2)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I2) ) ) ) ).

% transpose_column
tff(fact_4711_sorted__wrt__rev__linord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,aTP_Lamp_se(A,fun(A,bool)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_4712_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),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))) = rev(A,L) ) ) ).

% rev_split_conv
tff(fact_4713_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P,rev(A,Xs))
    <=> sorted_wrt(A,aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_4714_rev__butlast__is__tl__rev,axiom,
    ! [A: $tType,L: list(A)] : rev(A,butlast(A,L)) = aa(list(A),list(A),tl(A),rev(A,L)) ).

% rev_butlast_is_tl_rev
tff(fact_4715_butlast__rev__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( butlast(A,rev(A,Xs)) = rev(A,aa(list(A),list(A),tl(A),Xs)) ) ) ).

% butlast_rev_tl
tff(fact_4716_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_4717_rev__nth,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rev(A,Xs)),N2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,N2))) ) ) ).

% rev_nth
tff(fact_4718_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( rev(A,list_update(A,Xs,K,Y)) = list_update(A,rev(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_4719_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))),aa(nat,A,nth(A,Xs),I4))) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_4720_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I4))) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_4721_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I2))) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_4722_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( ( ( Xs = nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
        & ( ( Xs != nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).

% length_transpose_sorted
tff(fact_4723_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_vm(nat,fun(list(A),bool),I2)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) ) ) ) ).

% transpose_column_length
tff(fact_4724_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),X: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ~ pp(aa(set(B),bool,member(B,X),A5))
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A5)) = insort_key(A,B,Less_eq,F3,X,sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_4725_remove__rev__alt__def,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),remove_rev(A,X),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),aTP_Lamp_sp(A,fun(A,bool)),X)),rev(A,Xs)) ).

% remove_rev_alt_def
tff(fact_4726_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N2: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N2 = zero_zero(nat) ) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I3)) = N2 ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_vs(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N2)) ) ) ) ).

% transpose_rectangle
tff(fact_4727_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_si(list(A),bool),Xs) ) ) ).

% transpose_transpose
tff(fact_4728_sort__upt,axiom,
    ! [M: nat,N2: nat] : aa(list(nat),list(nat),linorder_sort_key(nat,nat,aTP_Lamp_ez(nat,nat)),upt(M,N2)) = upt(M,N2) ).

% sort_upt
tff(fact_4729_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_4730_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( aa(list(nat),nat,hd(nat),upt(I2,J)) = I2 ) ) ).

% hd_upt
tff(fact_4731_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( upt(I2,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_4732_upt__merge,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K)) )
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),upt(J,K)) = upt(I2,K) ) ) ).

% upt_merge
tff(fact_4733_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( upt(I2,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2)) ) ) ).

% upt_eq_Nil_conv
tff(fact_4734_take__upt,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)),N2))
     => ( take(nat,M,upt(I2,N2)) = upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)) ) ) ).

% take_upt
tff(fact_4735_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N2)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N2)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N2))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N2)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N2)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_4736_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( last(nat,upt(I2,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_4737_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J))
     => ( aa(nat,nat,nth(nat,upt(I2,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K) ) ) ).

% nth_upt
tff(fact_4738_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( groups8242544230860333062m_list(nat,upt(M,N2)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ez(nat,nat)),set_or7035219750837199246ssThan(nat,M,N2)) ) ) ).

% sum_list_upt
tff(fact_4739_map__add__upt_H,axiom,
    ! [Ofs: nat,A3: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_vt(nat,fun(nat,nat),Ofs)),upt(A3,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_4740_upt__eq__append__conv,axiom,
    ! [I2: nat,J: nat,Xs: list(nat),Ys2: list(nat)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( ( upt(I2,J) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xs),Ys2) )
      <=> ? [K3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),J))
            & ( upt(I2,K3) = Xs )
            & ( upt(K3,J) = Ys2 ) ) ) ) ).

% upt_eq_append_conv
tff(fact_4741_upt__append,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I2)),upt(I2,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_4742_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
       => ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
       => ( upt(I2,J) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_4743_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)) ) ) ).

% upt_conv_Cons
tff(fact_4744_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
       => ( upt(I2,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
       => ( upt(I2,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_4745_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( upt(I2,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_4746_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_4747_butlast__upt,axiom,
    ! [M: nat,N2: nat] : butlast(nat,upt(M,N2)) = upt(M,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) ).

% butlast_upt
tff(fact_4748_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_takeWhile_le
tff(fact_4749_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_4750_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : sorted_wrt(nat,ord_less(nat),upt(M,N2)) ).

% sorted_wrt_upt
tff(fact_4751_map__add__upt,axiom,
    ! [N2: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_vt(nat,fun(nat,nat),N2)),upt(zero_zero(nat),M)) = upt(N2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) ).

% map_add_upt
tff(fact_4752_sorted__upt,axiom,
    ! [M: nat,N2: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,N2)) ).

% sorted_upt
tff(fact_4753_filter__upt__take__conv,axiom,
    ! [A: $tType,P: fun(A,bool),M: nat,L: list(A),N2: nat] : aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_vu(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),P),M),L)),upt(N2,M)) = aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,bool),aTP_Lamp_vv(fun(A,bool),fun(list(A),fun(nat,bool)),P),L)),upt(N2,M)) ).

% filter_upt_take_conv
tff(fact_4754_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list(nat),I2: nat,Is2: list(nat)] :
      ( ( upt(L,H) = 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),I2),Is2)) )
    <=> ( ( Is1 = upt(L,I2) )
        & ( Is2 = upt(aa(nat,nat,suc,I2),H) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),I2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),H)) ) ) ).

% upt_eq_lel_conv
tff(fact_4755_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
        & ( I2 = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_4756_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
     => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))))) ) ).

% nth_length_takeWhile
tff(fact_4757_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_4758_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),U4))
     => ( ! [I3: nat] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),I3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),U4)) )
           => ~ pp(aa(nat,bool,P,I3)) )
       => ( aa(list(nat),list(nat),filter2(nat,P),upt(zero_zero(nat),U)) = aa(list(nat),list(nat),filter2(nat,P),upt(zero_zero(nat),U4)) ) ) ) ).

% upt_filter_extend
tff(fact_4759_map__decr__upt,axiom,
    ! [M: nat,N2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_pu(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N2))) = upt(M,N2) ).

% map_decr_upt
tff(fact_4760_drop__takeWhile,axiom,
    ! [A: $tType,I2: nat,P: fun(A,bool),L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L))))
     => ( drop(A,I2,takeWhile(A,P,L)) = takeWhile(A,P,drop(A,I2,L)) ) ) ).

% drop_takeWhile
tff(fact_4761_enumerate__map__upt,axiom,
    ! [A: $tType,N2: nat,F3: fun(nat,A),M: nat] : enumerate(A,N2,aa(list(nat),list(A),map(nat,A,F3),upt(N2,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_vw(fun(nat,A),fun(nat,product_prod(nat,A)),F3)),upt(N2,M)) ).

% enumerate_map_upt
tff(fact_4762_takeWhile__not__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( takeWhile(A,aTP_Lamp_vx(list(A),fun(A,bool),Xs),Xs) = butlast(A,Xs) ) ) ).

% takeWhile_not_last
tff(fact_4763_filter__upt__last,axiom,
    ! [A: $tType,P: fun(A,bool),L: list(A),Js: list(nat),J: nat,I2: nat] :
      ( ( aa(list(nat),list(nat),filter2(nat,aa(list(A),fun(nat,bool),aTP_Lamp_vv(fun(A,bool),fun(list(A),fun(nat,bool)),P),L)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
         => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,L),I2))) ) ) ) ).

% filter_upt_last
tff(fact_4764_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_4765_eq__len__takeWhile__conv,axiom,
    ! [A: $tType,I2: nat,P: fun(A,bool),L: list(A)] :
      ( ( I2 = aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
        & ! [J3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I2))
           => pp(aa(A,bool,P,aa(nat,A,nth(A,L),J3))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
         => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,L),I2))) ) ) ) ).

% eq_len_takeWhile_conv
tff(fact_4766_less__length__takeWhile__conv,axiom,
    ! [A: $tType,I2: nat,P: fun(A,bool),L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
        & ! [J3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J3),I2))
           => pp(aa(A,bool,P,aa(nat,A,nth(A,L),J3))) ) ) ) ).

% less_length_takeWhile_conv
tff(fact_4767_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),P: fun(A,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
         => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N2))) )
       => ( takeWhile(A,P,Xs) = take(A,N2,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_4768_map__upt__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),N2: nat] : aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),aa(nat,nat,suc,N2))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F3,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_vy(fun(nat,A),fun(nat,A),F3)),upt(zero_zero(nat),N2))) ).

% map_upt_Suc
tff(fact_4769_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(nat),list(A),map(nat,A,nth(A,Xs)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_4770_nth__map__upt,axiom,
    ! [A: $tType,I2: nat,N2: nat,M: nat,F3: fun(nat,A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F3),upt(M,N2))),I2) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) ) ).

% nth_map_upt
tff(fact_4771_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N2: nat,M: nat,F3: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I3)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F3),upt(M,N2)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_4772_map__nth__upt__drop__take__conv,axiom,
    ! [A: $tType,N7: nat,L: list(A),M6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),aa(list(A),nat,size_size(list(A)),L)))
     => ( aa(list(nat),list(A),map(nat,A,nth(A,L)),upt(M6,N7)) = drop(A,M6,take(A,N7,L)) ) ) ).

% map_nth_upt_drop_take_conv
tff(fact_4773_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),T6: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(B),list(A),map(B,A,F3),Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,bool),aTP_Lamp_ty(fun(B,A),fun(A,fun(B,bool)),F3),T6)),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_ty(fun(B,A),fun(A,fun(B,bool)),F3),T6),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_4774_extract__def,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_vz(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P),Xs)) ).

% extract_def
tff(fact_4775_map__filter__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F3,Xs) = aa(list(A),list(B),map(A,B,aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),F3)),aa(list(A),list(A),filter2(A,aTP_Lamp_wa(fun(A,option(B)),fun(A,bool),F3)),Xs)) ).

% map_filter_def
tff(fact_4776_remove__rev__def,axiom,
    ! [A: $tType,X: A] : remove_rev(A,X) = aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),aa(A,fun(A,bool),fequal(A),X))) ).

% remove_rev_def
tff(fact_4777_option_Ocollapse,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) = Option ) ) ).

% option.collapse
tff(fact_4778_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),Q: fun(B,bool),G3: fun(A,B),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,aa(fun(B,bool),fun(B,bool),aTP_Lamp_wb(fun(B,bool),fun(fun(B,bool),fun(B,bool)),P),Q)),G3),X5))
    <=> ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,P),G3),X5))
        & pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,Q),G3),X5)) ) ) ).

% conj_comp_iff
tff(fact_4779_option_Osel,axiom,
    ! [A: $tType,X2: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X2)) = X2 ).

% option.sel
tff(fact_4780_option_Oexpand,axiom,
    ! [A: $tType,Option: option(A),Option2: option(A)] :
      ( ( ( Option = none(A) )
      <=> ( Option2 = none(A) ) )
     => ( ( ( Option != none(A) )
         => ( ( Option2 != none(A) )
           => ( aa(option(A),A,the2(A),Option) = aa(option(A),A,the2(A),Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4781_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_dropWhile_le
tff(fact_4782_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_4783_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) ) ) ).

% option.exhaust_sel
tff(fact_4784_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A3: option(A),F3: fun(A,B)] :
      ( ( A3 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),map_option(A,B,F3),A3)) = aa(A,B,F3,aa(option(A),A,the2(A),A3)) ) ) ).

% option.map_sel
tff(fact_4785_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F1: B,F22: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( case_option(B,A,F1,F22,Option) = F1 ) )
      & ( ( Option != none(A) )
       => ( case_option(B,A,F1,F22,Option) = aa(A,B,F22,aa(option(A),A,the2(A),Option)) ) ) ) ).

% option.case_eq_if
tff(fact_4786_remdups__adj__Cons_H,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups_adj(A,dropWhile(A,aTP_Lamp_br(A,fun(A,bool),X),Xs))) ).

% remdups_adj_Cons'
tff(fact_4787_Option_Othese__def,axiom,
    ! [A: $tType,A5: set(option(A))] : these(A,A5) = aa(set(option(A)),set(A),image2(option(A),A,the2(A)),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_pv(set(option(A)),fun(option(A),bool),A5))) ).

% Option.these_def
tff(fact_4788_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_4789_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F1,F22,Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,P,F1)) )
        & ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
         => pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel
tff(fact_4790_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F1,F22,Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,P,F1)) )
          | ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
            & ~ pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel_asm
tff(fact_4791_length__dropWhile__takeWhile,axiom,
    ! [A: $tType,X: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_dropWhile_takeWhile
tff(fact_4792_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys2: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_br(A,fun(A,bool),Y),Ys2))) ).

% remdups_adj_append_dropWhile
tff(fact_4793_remdups__adj__append_H_H,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != nil(A) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,dropWhile(A,aTP_Lamp_wc(list(A),fun(A,bool),Xs),Ys2))) ) ) ).

% remdups_adj_append''
tff(fact_4794_tl__remdups__adj,axiom,
    ! [A: $tType,Ys2: list(A)] :
      ( ( Ys2 != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys2)) = remdups_adj(A,dropWhile(A,aTP_Lamp_wd(list(A),fun(A,bool),Ys2),aa(list(A),list(A),tl(A),Ys2))) ) ) ).

% tl_remdups_adj
tff(fact_4795_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))))
     => ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_4796_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ( dropWhile(A,aa(A,fun(A,bool),aTP_Lamp_sp(A,fun(A,bool)),X),rev(A,Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),rev(A,takeWhile(A,aa(A,fun(A,bool),aTP_Lamp_sp(A,fun(A,bool)),X),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_4797_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ( takeWhile(A,aa(A,fun(A,bool),aTP_Lamp_sp(A,fun(A,bool)),X),rev(A,Xs)) = rev(A,aa(list(A),list(A),tl(A),dropWhile(A,aa(A,fun(A,bool),aTP_Lamp_sp(A,fun(A,bool)),X),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_4798_filter__rev__alt,axiom,
    ! [A: $tType,P: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),P),L) = aa(list(A),list(A),filter2(A,P),rev(A,L)) ).

% filter_rev_alt
tff(fact_4799_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_we(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P),Xs)) ).

% find_dropWhile
tff(fact_4800_partition__filter__conv,axiom,
    ! [A: $tType,F3: fun(A,bool),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F3),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),filter2(A,F3),Xs)),aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),F3)),Xs)) ).

% partition_filter_conv
tff(fact_4801_partition__rev__filter__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Yes2: list(A),No2: list(A),Xs: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,aa(list(A),list(A),filter2(A,P),Xs))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P)),Xs))),No2)) ).

% partition_rev_filter_conv
tff(fact_4802_find__SomeD_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
     => pp(aa(A,bool,P,X)) ) ).

% find_SomeD(1)
tff(fact_4803_partition__rev_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),Yes2: list(A),No2: list(A),X: A,Xs: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = partition_rev(A,P,if(product_prod(list(A),list(A)),aa(A,bool,P,X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Yes2)),No2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),No2))),Xs) ).

% partition_rev.simps(2)
tff(fact_4804_partition__rev_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),Yes2: list(A),No2: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) ).

% partition_rev.simps(1)
tff(fact_4805_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,option(A),some(A),X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = find(A,P,Xs) ) ) ) ).

% find.simps(2)
tff(fact_4806_find__SomeD_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ).

% find_SomeD(2)
tff(fact_4807_partition_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)) ).

% partition.simps(1)
tff(fact_4808_partition__P,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes2: list(A),No2: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
     => ( ! [X5: A] :
            ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Yes2)))
           => pp(aa(A,bool,P,X5)) )
        & ! [X5: A] :
            ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),No2)))
           => ~ pp(aa(A,bool,P,X5)) ) ) ) ).

% partition_P
tff(fact_4809_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_wf(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),X)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) ).

% partition.simps(2)
tff(fact_4810_partition__rev_Oelims,axiom,
    ! [A: $tType,X: fun(A,bool),Xa2: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,X,Xa2,Xb) = Y )
     => ( ! [Yes: list(A),No: list(A)] :
            ( ( Xa2 = aa(list(A),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)] :
              ( ( Xa2 = aa(list(A),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) )
             => ! [X3: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( Y != partition_rev(A,X,if(product_prod(list(A),list(A)),aa(A,bool,X,X3),aa(list(A),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),X3),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),X3),No))),Xs2) ) ) ) ) ) ).

% partition_rev.elims
tff(fact_4811_partition__set,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes2: list(A),No2: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes2)),aa(list(A),set(A),set2(A),No2)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_4812_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4)))
          & ( X = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I4))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_4813_find__Some__iff2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),X) = find(A,P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4)))
          & ( X = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I4))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_4814_partition__rev_Opelims,axiom,
    ! [A: $tType,X: fun(A,bool),Xa2: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,X,Xa2,Xb) = Y )
     => ( pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),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,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xa2),Xb))))
       => ( ! [Yes: list(A),No: list(A)] :
              ( ( Xa2 = aa(list(A),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) )
                 => ~ pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),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,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A))))) ) ) )
         => ~ ! [Yes: list(A),No: list(A)] :
                ( ( Xa2 = aa(list(A),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) )
               => ! [X3: A,Xs2: list(A)] :
                    ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ( ( Y = partition_rev(A,X,if(product_prod(list(A),list(A)),aa(A,bool,X,X3),aa(list(A),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),X3),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),X3),No))),Xs2) )
                     => ~ pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),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,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))))) ) ) ) ) ) ) ).

% partition_rev.pelims
tff(fact_4815_quicksort__by__rel_Osimps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),X: A,Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_wg(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R2),Sl2),X)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R2),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ).

% quicksort_by_rel.simps(2)
tff(fact_4816_quicksort__by__rel_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa2),Xb) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != Xa2 ) )
       => ~ ! [X3: A,Xs2: list(A)] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ( 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_wg(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa2),X3)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X),X3),aa(list(A),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)),Xs2)) ) ) ) ) ).

% quicksort_by_rel.elims
tff(fact_4817_set__quicksort__by__rel,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).

% set_quicksort_by_rel
tff(fact_4818_quicksort__by__rel_Osimps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),nil(A)) = Sl2 ).

% quicksort_by_rel.simps(1)
tff(fact_4819_quicksort__by__rel__remove__acc__guared,axiom,
    ! [A: $tType,Sl2: list(A),R2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( ( Sl2 != nil(A) )
     => ( aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R2,nil(A)),Xs)),Sl2) ) ) ).

% quicksort_by_rel_remove_acc_guared
tff(fact_4820_quicksort__by__rel__remove__acc,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R2,nil(A)),Xs)),Sl2) ).

% quicksort_by_rel_remove_acc
tff(fact_4821_sorted__wrt__quicksort__by__rel,axiom,
    ! [X10: $tType,R2: fun(X10,fun(X10,bool)),Xs: list(X10)] :
      ( ! [X3: X10,Y3: X10] :
          ( pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,X3),Y3))
          | pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,Y3),X3)) )
     => ( ! [X3: X10,Y3: X10,Z3: X10] :
            ( pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,X3),Y3))
           => ( pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,Y3),Z3))
             => pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,X3),Z3)) ) )
       => sorted_wrt(X10,R2,aa(list(X10),list(X10),quicksort_by_rel(X10,R2,nil(X10)),Xs)) ) ) ).

% sorted_wrt_quicksort_by_rel
tff(fact_4822_sorted__quicksort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),quicksort_by_rel(A,ord_less_eq(A),nil(A)),Xs)) ) ).

% sorted_quicksort_by_rel
tff(fact_4823_sort__quicksort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_rm(A,A)) = quicksort_by_rel(A,ord_less_eq(A),nil(A)) ) ) ).

% sort_quicksort_by_rel
tff(fact_4824_quicksort__by__rel_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa2),Xb) = Y )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa2),Xb))))
       => ( ( ( Xb = nil(A) )
           => ( ( Y = Xa2 )
             => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa2),nil(A))))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( ( 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_wg(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa2),X3)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X),X3),aa(list(A),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)),Xs2)) )
                 => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))))) ) ) ) ) ) ).

% quicksort_by_rel.pelims
tff(fact_4825_quicksort__by__rel_Opsimps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),X: A,Xs: list(A)] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))))
     => ( aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_wg(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R2),Sl2),X)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R2),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ) ) ).

% quicksort_by_rel.psimps(2)
tff(fact_4826_quicksort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),A22: list(A),P: fun(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)))] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),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))))
     => ( ! [R9: fun(A,fun(A,bool)),Sl: list(A)] :
            ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R9),aa(list(A),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)))))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R9),Sl),nil(A))) )
       => ( ! [R9: fun(A,fun(A,bool)),Sl: list(A),X3: A,Xs2: list(A)] :
              ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R9),aa(list(A),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),X3),Xs2)))))
             => ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y4: list(A)] :
                    ( ( Xa3 = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R9),X3),aa(list(A),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)),Xs2) )
                   => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y4) = Xa3 )
                     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R9),Sl),Y4)) ) )
               => ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y4: list(A)] :
                      ( ( Xa3 = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R9),X3),aa(list(A),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)),Xs2) )
                     => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y4) = Xa3 )
                       => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R9),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),quicksort_by_rel(A,R9,Sl),Y4))),Xb2)) ) )
                 => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R9),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ) )
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,A0),A1),A22)) ) ) ) ).

% quicksort_by_rel.pinduct
tff(fact_4827_quicksort__by__rel_Opsimps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A)] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R2),aa(list(A),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,R2,Sl2),nil(A)) = Sl2 ) ) ).

% quicksort_by_rel.psimps(1)
tff(fact_4828_filter__rev__aux__alt,axiom,
    ! [A: $tType,A3: list(A),P: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,P),rev(A,L))),A3) ).

% filter_rev_aux_alt
tff(fact_4829_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3)))
    <=> ? [Y5: A,N: nat] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N)),Y5)),R3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( Ys2 = list_update(A,Xs,N,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_4830_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_wh(fun(A,B),fun(A,product_prod(A,B)),F3)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F3),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_4831_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_bk(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_4832_Cons__listrel1__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),listrel1(A,R3)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
          & ( Xs = Ys2 ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4833_listrel1__mono,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel1(A,S2))) ) ).

% listrel1_mono
tff(fact_4834_map__of__None__filterD,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),X: B,P: fun(product_prod(B,A),bool)] :
      ( ( aa(B,option(A),map_of(B,A,Xs),X) = none(A) )
     => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),filter2(product_prod(B,A),P),Xs)),X) = none(A) ) ) ).

% map_of_None_filterD
tff(fact_4835_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,Z2: A,P: fun(B,fun(A,bool))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Z2) )
     => ( pp(aa(A,bool,aa(B,fun(A,bool),P,K),Z2))
       => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),filter2(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P)),Xs)),K) = aa(A,option(A),some(A),Z2) ) ) ) ).

% map_of_filter_in
tff(fact_4836_listrel1I2,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A)),X: A] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))),listrel1(A,R3))) ) ).

% listrel1I2
tff(fact_4837_not__listrel1__Nil,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))),listrel1(A,R3))) ).

% not_listrel1_Nil
tff(fact_4838_not__Nil__listrel1,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xs)),listrel1(A,R3))) ).

% not_Nil_listrel1
tff(fact_4839_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) ) ) ).

% listrel1_eq_len
tff(fact_4840_append__listrel1I,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
      ( ( ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3)))
          & ( Us = Vs ) )
        | ( ( Xs = Ys2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Vs)),listrel1(A,R3))) ) )
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Vs))),listrel1(A,R3))) ) ).

% append_listrel1I
tff(fact_4841_filter__rev__aux_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,A3: list(A),Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A3)),P),Xs) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),Xs) ) ) ) ).

% filter_rev_aux.simps(2)
tff(fact_4842_filter__rev__aux_Osimps_I1_J,axiom,
    ! [A: $tType,A3: list(A),P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),nil(A)) = A3 ).

% filter_rev_aux.simps(1)
tff(fact_4843_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K: A,X: B,L: list(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),X)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L)))
     => ? [X3: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X3) ) ).

% weak_map_of_SomeI
tff(fact_4844_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ) ).

% map_of_SomeD
tff(fact_4845_map__of__Cons__code_I2_J,axiom,
    ! [C: $tType,B: $tType,L: B,K: B,V2: C,Ps: list(product_prod(B,C))] :
      ( ( ( L = K )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V2)),Ps)),K) = aa(C,option(C),some(C),V2) ) )
      & ( ( L != K )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V2)),Ps)),K) = aa(B,option(C),map_of(B,C,Ps),K) ) ) ) ).

% map_of_Cons_code(2)
tff(fact_4846_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),listrel1(A,R3)))
     => ( ! [X3: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2) )
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y)),R3)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1E2
tff(fact_4847_Cons__listrel1E1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys2)),listrel1(A,R3)))
     => ( ! [Y3: A] :
            ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs) )
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R3)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1E1
tff(fact_4848_listrel1I1,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))),listrel1(A,R3))) ) ).

% listrel1I1
tff(fact_4849_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3)))
     => ~ ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
           => ! [Us2: list(A),Vs2: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Vs2)) )
               => ( Ys2 != 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),Y3),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_4850_listrel1I,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys2: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
       => ( ( Ys2 = 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)) )
         => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3))) ) ) ) ).

% listrel1I
tff(fact_4851_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_wi(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_4852_map__of__Some__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V2: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V2) )
     => ? [Ys3: list(product_prod(B,A)),Zs2: list(product_prod(B,A))] :
          ( ( Xs = aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(list(product_prod(B,A)),fun(list(product_prod(B,A)),list(product_prod(B,A))),append(product_prod(B,A)),Ys3),aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V2)),Zs2)) )
          & ( aa(B,option(A),map_of(B,A,Ys3),K) = none(A) ) ) ) ).

% map_of_Some_split
tff(fact_4853_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys2: list(A),Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),listrel1(A,R3)))
    <=> ( ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3)))
          & ( X = Y ) )
        | ( ( Xs = Ys2 )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_4854_filter__rev__def,axiom,
    ! [A: $tType] : filter_rev(A) = filter_rev_aux(A,nil(A)) ).

% filter_rev_def
tff(fact_4855_listrel1p__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] :
      ( listrel1p(A,R3,Xs,Ys2)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% listrel1p_def
tff(fact_4856_map__of__distinct__upd4,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys2)) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys2))),X,none(B)) ) ) ) ).

% map_of_distinct_upd4
tff(fact_4857_map__of__distinct__upd3,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B,Y7: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys2))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),Ys2))),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd3
tff(fact_4858_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_4859_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),X: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),X)) = aa(product_prod(A,C),A,product_fst(A,C),X) ).

% fst_apsnd
tff(fact_4860_img__fst,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
     => pp(aa(set(A),bool,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_4861_map__of__rev__distinct,axiom,
    ! [B: $tType,A: $tType,M: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),M))
     => ( map_of(A,B,rev(product_prod(A,B),M)) = map_of(A,B,M) ) ) ).

% map_of_rev_distinct
tff(fact_4862_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_4863_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_wj(product_prod(A,B),fun(product_prod(A,B),bool)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L)) ) ) ).

% sorted_wrt_map_linord
tff(fact_4864_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_wk(B,fun(A,product_prod(A,B))),K)),L)) = L ).

% map_fst_mk_snd
tff(fact_4865_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_wl(product_prod(A,B),fun(product_prod(A,B),bool)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))) ) ) ).

% sorted_wrt_map_rev_linord
tff(fact_4866_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)))
       => ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_4867_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X) )
      <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).

% Some_eq_map_of_iff
tff(fact_4868_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) )
      <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).

% map_of_eq_Some_iff
tff(fact_4869_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_uv(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_4870_fst__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_px(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_4871_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] : aTP_Lamp_wm(fun(A,C),fun(product_prod(A,B),C),F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_wn(fun(A,C),fun(A,fun(B,C)),F3)) ).

% fn_fst_conv
tff(fact_4872_distinct__map__fst__filterI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),P: fun(product_prod(A,B),bool)] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => distinct(A,aa(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)),filter2(product_prod(A,B),P),Xs))) ) ).

% distinct_map_fst_filterI
tff(fact_4873_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X2: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2)) = X1 ).

% fst_conv
tff(fact_4874_fst__eqD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
tff(fact_4875_fstE,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(A,bool)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
     => ( pp(aa(A,bool,P,aa(product_prod(A,B),A,product_fst(A,B),X)))
       => pp(aa(A,bool,P,A3)) ) ) ).

% fstE
tff(fact_4876_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,X: A,S: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)))
     => ~ ! [Y3: B] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)),S)) ) ).

% in_fst_imageE
tff(fact_4877_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(A,B)),B4: set(A),X: A,Y: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A5)),B4))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),A5))
       => pp(aa(set(A),bool,member(A,X),B4)) ) ) ).

% fst_image_mp
tff(fact_4878_distinct__map__fstD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),X: A,Y: B,Z2: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
         => ( Y = Z2 ) ) ) ) ).

% distinct_map_fstD
tff(fact_4879_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V1)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V22)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_4880_sorted__enumerate,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,N2,Xs))) ).

% sorted_enumerate
tff(fact_4881_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),M,K,none(B))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),bool),aTP_Lamp_wo(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M),K)) ).

% graph_fun_upd_None
tff(fact_4882_fst__foldl,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,fun(C,A)),G3: fun(A,fun(B,fun(C,B))),A3: A,B2: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(list(C),product_prod(A,B),aa(product_prod(A,B),fun(list(C),product_prod(A,B)),foldl(product_prod(A,B),C,aa(fun(A,fun(B,fun(C,product_prod(A,B)))),fun(product_prod(A,B),fun(C,product_prod(A,B))),product_case_prod(A,B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_wp(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F3),G3))),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),foldl(A,C,F3),A3),Xs) ).

% fst_foldl
tff(fact_4883_map__of__map__to__set,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B)),M: fun(A,option(B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
     => ( ( map_of(A,B,L) = M )
      <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) = map_to_set(A,B,M) ) ) ) ).

% map_of_map_to_set
tff(fact_4884_map__to__set__map__of,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),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_4885_map__of__Some__filter__not__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V2: A,P: fun(product_prod(B,A),bool)] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V2) )
     => ( ~ pp(aa(product_prod(B,A),bool,P,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V2)))
       => ( distinct(B,aa(list(product_prod(B,A)),list(B),map(product_prod(B,A),B,product_fst(B,A)),Xs))
         => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),filter2(product_prod(B,A),P),Xs)),K) = none(A) ) ) ) ) ).

% map_of_Some_filter_not_in
tff(fact_4886_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_wq(list(product_prod(A,B)),fun(A,fun(B,bool)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_4887_map__of__distinct__lookup,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( aa(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys2))),X) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_distinct_lookup
tff(fact_4888_map__of__distinct__upd2,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys2))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys2)),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd2
tff(fact_4889_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
        & ( ( Xa2 != zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_4890_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero(nat) )
       => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
      & ( ( Y != zero_zero(nat) )
       => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_4891_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa2))))) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))) ) ) ) ).

% bezw.pelims
tff(fact_4892_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),X: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F3),X)) = aa(C,A,F3,aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% snd_apsnd
tff(fact_4893_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),X: product_prod(A,C),G3: fun(C,B)] :
      ( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),X) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G3),X) )
    <=> ( aa(C,B,F3,aa(product_prod(A,C),C,product_snd(A,C),X)) = aa(C,B,G3,aa(product_prod(A,C),C,product_snd(A,C),X)) ) ) ).

% apsnd_eq_conv
tff(fact_4894_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_4895_img__snd,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
     => pp(aa(set(B),bool,member(B,B2),aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S))) ) ).

% img_snd
tff(fact_4896_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_4897_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(B,A)),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K)),L)) = L ).

% map_snd_mk_fst
tff(fact_4898_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),F3),product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_4899_prod_Oexpand,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
      ( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_4900_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),Q3: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P3) = aa(product_prod(A,B),A,product_fst(A,B),Q3) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P3) = aa(product_prod(A,B),B,product_snd(A,B),Q3) )
       => ( P3 = Q3 ) ) ) ).

% prod_eqI
tff(fact_4901_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S2: product_prod(A,B),T6: product_prod(A,B)] :
      ( ( S2 = T6 )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S2) = aa(product_prod(A,B),A,product_fst(A,B),T6) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S2) = aa(product_prod(A,B),B,product_snd(A,B),T6) ) ) ) ).

% prod_eq_iff
tff(fact_4902_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
      ( ? [A8: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A8),X_12))
    <=> ? [Z6: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(A,B),A,product_fst(A,B),Z6)),aa(product_prod(A,B),B,product_snd(A,B),Z6))) ) ).

% Ex_prod_contract
tff(fact_4903_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
      ( ! [A8: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A8),X_12))
    <=> ! [Z6: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(A,B),A,product_fst(A,B),Z6)),aa(product_prod(A,B),B,product_snd(A,B),Z6))) ) ).

% All_prod_contract
tff(fact_4904_sndE,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(B,bool)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
     => ( pp(aa(B,bool,P,aa(product_prod(A,B),B,product_snd(A,B),X)))
       => pp(aa(B,bool,P,B2)) ) ) ).

% sndE
tff(fact_4905_snd__eqD,axiom,
    ! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
      ( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
tff(fact_4906_snd__conv,axiom,
    ! [Aa: $tType,A: $tType,X1: Aa,X2: A] : aa(product_prod(Aa,A),A,product_snd(Aa,A),aa(A,product_prod(Aa,A),aa(Aa,fun(A,product_prod(Aa,A)),product_Pair(Aa,A),X1),X2)) = X2 ).

% snd_conv
tff(fact_4907_fn__snd__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] : aTP_Lamp_wr(fun(B,C),fun(product_prod(A,B),C),F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_ws(fun(B,C),fun(A,fun(B,C)),F3)) ).

% fn_snd_conv
tff(fact_4908_snd__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,aa(fun(A,fun(B,B)),fun(product_prod(A,B),B),product_case_prod(A,B,B),aTP_Lamp_wt(A,fun(B,B))),Prod) ).

% snd_def
tff(fact_4909_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_wu(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_4910_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Y: A,X: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,Y),X))
     => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)))) ) ).

% exI_realizer
tff(fact_4911_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),P3: A,Q: fun(B,bool),Q3: B] :
      ( pp(aa(A,bool,P,P3))
     => ( pp(aa(B,bool,Q,Q3))
       => ( pp(aa(A,bool,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3))))
          & pp(aa(B,bool,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3)))) ) ) ) ).

% conjI_realizer
tff(fact_4912_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T6: product_prod(A,B)] : T6 = 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),T6)),aa(product_prod(A,B),B,product_snd(A,B),T6)) ).

% surjective_pairing
tff(fact_4913_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_4914_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),X: A,Y: B,A3: product_prod(A,B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
     => ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3))) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4915_split__beta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% split_beta
tff(fact_4916_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),P3: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),P3) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),P3)),aa(product_prod(B,C),C,product_snd(B,C),P3)) ).

% case_prod_beta
tff(fact_4917_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),A5: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A5))))
     => pp(aa(B,bool,aa(A,fun(B,bool),A5,aa(product_prod(A,B),A,product_fst(A,B),X)),aa(product_prod(A,B),B,product_snd(A,B),X))) ) ).

% Product_Type.Collect_case_prodD
tff(fact_4918_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,S: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,member(A,Y),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S)))
     => ~ ! [X3: B] : ~ pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y)),S)) ) ).

% in_snd_imageE
tff(fact_4919_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_wu(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_4920_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_uv(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_4921_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),P3: product_prod(B,A),Q: fun(C,bool),F3: fun(B,fun(A,C))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(B,A),A,product_snd(B,A),P3)),aa(product_prod(B,A),B,product_fst(B,A),P3)))
     => ( ! [X3: B,Y3: A] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),P,Y3),X3))
           => pp(aa(C,bool,Q,aa(A,C,aa(B,fun(A,C),F3,X3),Y3))) )
       => pp(aa(C,bool,Q,aa(product_prod(B,A),C,aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),F3),P3))) ) ) ).

% exE_realizer
tff(fact_4922_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,F3: fun(A,fun(B,C)),G3: fun(D,A)] : aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_wv(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),F3),G3) = aa(fun(D,fun(B,C)),fun(product_prod(D,B),C),product_case_prod(D,B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_ww(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),F3),G3)) ).

% split_comp_eq
tff(fact_4923_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),X5: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),X5) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),X5)),aa(product_prod(A,B),B,product_snd(A,B),X5)) ).

% case_prod_beta'
tff(fact_4924_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X5: fun(A,fun(B,C)),Xa3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X5),Xa3) = aa(B,C,aa(A,fun(B,C),X5,aa(product_prod(A,B),A,product_fst(A,B),Xa3)),aa(product_prod(A,B),B,product_snd(A,B),Xa3)) ).

% case_prod_unfold
tff(fact_4925_prod_Osplit__sel__asm,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)))
    <=> ~ ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
          & ~ pp(aa(C,bool,P,aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel_asm
tff(fact_4926_prod_Osplit__sel,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)))
    <=> ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
       => pp(aa(C,bool,P,aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel
tff(fact_4927_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(B,A)),B4: set(A),X: B,Y: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A5)),B4))
     => ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)),A5))
       => pp(aa(set(A),bool,member(A,Y),B4)) ) ) ).

% snd_image_mp
tff(fact_4928_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(D,fun(C,A)),G3: fun(B,D),X: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(fun(B,D),fun(B,fun(C,A)),comp(D,fun(C,A),B,F3),G3)),X) = aa(C,A,aa(D,fun(C,A),F3,aa(B,D,G3,aa(product_prod(B,C),B,product_fst(B,C),X))),aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% case_prod_comp
tff(fact_4929_map__to__set__ran,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),map_to_set(B,A,M)) ).

% map_to_set_ran
tff(fact_4930_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X6: set(A),A5: set(product_prod(A,B)),Y6: set(B),P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
      ( ( X6 = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A5) )
     => ( ( Y6 = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),A5) )
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => ! [Xa4: B] :
                  ( pp(aa(set(B),bool,member(B,Xa4),Y6))
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Xa4))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Xa4)) ) ) )
         => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P))))
           => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q)))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_4931_set__to__map__ran,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_4932_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,product_prod(A,C))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,aa(fun(B,product_prod(A,C)),fun(B,A),comp(product_prod(A,C),A,B,product_fst(A,C)),F3)),top_top(set(B)))))
     => ( pp(aa(set(C),bool,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(B,product_prod(A,C)),fun(B,C),comp(product_prod(A,C),C,B,product_snd(A,C)),F3)),top_top(set(B)))))
       => pp(aa(set(product_prod(A,C)),bool,finite_finite2(product_prod(A,C)),aa(set(B),set(product_prod(A,C)),image2(B,product_prod(A,C),F3),top_top(set(B))))) ) ) ).

% finite_range_prod
tff(fact_4933_ran__map__of,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ran(B,A,map_of(B,A,Xs))),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)),Xs)))) ).

% ran_map_of
tff(fact_4934_map__of_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),Ps: list(product_prod(A,B))] : map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),P3),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P3),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P3))) ).

% map_of.simps(2)
tff(fact_4935_set__to__map__insert,axiom,
    ! [B: $tType,A: $tType,Kv: product_prod(A,B),S: set(product_prod(A,B))] :
      ( ~ pp(aa(set(A),bool,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))),insert(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_4936_Misc_Oran__distinct,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),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_4937_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
     => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ).

% bezw_non_0
tff(fact_4938_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_wx(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_4939_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_wk(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_4940_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,nat),P3: product_prod(A,B)] : basic_BNF_size_prod(A,B,F3,G3,P3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(product_prod(A,B),A,product_fst(A,B),P3))),aa(B,nat,G3,aa(product_prod(A,B),B,product_snd(A,B),P3)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_4941_mod__pure,axiom,
    ! [B2: bool,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(pure_assn(B2)),H))
    <=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) )
        & pp(B2) ) ) ).

% mod_pure
tff(fact_4942_effectE,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,N2: nat] :
      ( heap_Time_effect(A,C2,H,H3,R3,N2)
     => ~ ( ( R3 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,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,C2),H))) )
         => ( ( H3 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,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,C2),H)))) )
           => ( ( N2 = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,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,C2),H)))) )
             => ~ heap_Time_success(A,C2,H) ) ) ) ) ).

% effectE
tff(fact_4943_execute__bind__success,axiom,
    ! [B: $tType,A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),G3: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_success(A,F3,H)
     => ( 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,F3,G3)),H) = heap_Time_timeFrame(B,aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,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,F3),H)))),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),G3,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,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,F3),H))))),aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,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,F3),H)))))) ) ) ).

% execute_bind_success
tff(fact_4944_mod__emp,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(one_one(assn)),H))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) ) ) ).

% mod_emp
tff(fact_4945_ge__eq__refl,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R2))
     => pp(aa(A,bool,aa(A,fun(A,bool),R2,X),X)) ) ).

% ge_eq_refl
tff(fact_4946_refl__ge__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( ! [X3: A] : pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),X3))
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R2)) ) ).

% refl_ge_eq
tff(fact_4947_mod__star__trueE_H,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H))
     => ~ ! [H5: 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)),H5) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H) )
           => ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H5)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H)))
             => ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),H5)) ) ) ) ).

% mod_star_trueE'
tff(fact_4948_fstI,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),Y: A,Z2: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),X) = Y ) ) ).

% fstI
tff(fact_4949_sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),Y: A,Z2: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),X) = Z2 ) ) ).

% sndI
tff(fact_4950_in__set__enumerate__eq,axiom,
    ! [A: $tType,P3: product_prod(nat,A),N2: nat,Xs: list(A)] :
      ( pp(aa(set(product_prod(nat,A)),bool,member(product_prod(nat,A),P3),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N2,Xs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N2)))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),N2)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).

% in_set_enumerate_eq
tff(fact_4951_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs12)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),modulo_modulo(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
      & ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs23)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_4952_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_wz(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_4953_relcomp__empty2,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(A,C))] : relcomp(A,C,B,R2,bot_bot(set(product_prod(C,B)))) = bot_bot(set(product_prod(A,B))) ).

% relcomp_empty2
tff(fact_4954_relcomp__empty1,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(C,B))] : relcomp(A,C,B,bot_bot(set(product_prod(A,C))),R2) = bot_bot(set(product_prod(A,B))) ).

% relcomp_empty1
tff(fact_4955_relcomp__distrib,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S: set(product_prod(C,B)),T2: set(product_prod(C,B))] : relcomp(A,C,B,R2,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),S),T2)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),relcomp(A,C,B,R2,S)),relcomp(A,C,B,R2,T2)) ).

% relcomp_distrib
tff(fact_4956_relcomp__distrib2,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: set(product_prod(A,C)),T2: set(product_prod(A,C)),R2: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),set(product_prod(A,C))),sup_sup(set(product_prod(A,C))),S),T2),R2) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),relcomp(A,C,B,S,R2)),relcomp(A,C,B,T2,R2)) ).

% relcomp_distrib2
tff(fact_4957_relcomp__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,R6: set(product_prod(A,B)),R3: set(product_prod(A,B)),S4: set(product_prod(B,C)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R6),R3))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),bool),ord_less_eq(set(product_prod(B,C))),S4),S2))
       => pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R6,S4)),relcomp(A,B,C,R3,S2))) ) ) ).

% relcomp_mono
tff(fact_4958_O__assoc,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,R2: set(product_prod(A,D)),S: set(product_prod(D,C)),T2: set(product_prod(C,B))] : relcomp(A,C,B,relcomp(A,D,C,R2,S),T2) = relcomp(A,D,B,R2,relcomp(D,C,B,S,T2)) ).

% O_assoc
tff(fact_4959_relcomp_Ocases,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R3: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R3,S2)))
     => ~ ! [B5: B] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B5)),R3))
           => ~ pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B5),A22)),S2)) ) ) ).

% relcomp.cases
tff(fact_4960_relcomp_Osimps,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R3: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R3,S2)))
    <=> ? [A8: A,B13: B,C5: C] :
          ( ( A1 = A8 )
          & ( A22 = C5 )
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B13)),R3))
          & pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B13),C5)),S2)) ) ) ).

% relcomp.simps
tff(fact_4961_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R3: set(product_prod(A,B)),C2: C,S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
     => ( pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2)),S2))
       => pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2)),relcomp(A,B,C,R3,S2))) ) ) ).

% relcomp.relcompI
tff(fact_4962_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R3: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Xz),relcomp(A,C,B,R3,S2)))
     => ~ ! [X3: A,Y3: C,Z3: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3) )
           => ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X3),Y3)),R3))
             => ~ pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Z3)),S2)) ) ) ) ).

% relcompE
tff(fact_4963_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R3: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),C2)),relcomp(A,C,B,R3,S2)))
     => ~ ! [B5: C] :
            ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B5)),R3))
           => ~ pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B5),C2)),S2)) ) ) ).

% relcompEpair
tff(fact_4964_union__comp__emptyL,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),C4: set(product_prod(A,A)),B4: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A5,C4) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,B4,C4) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A5),B4),C4) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyL
tff(fact_4965_union__comp__emptyR,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),B4: set(product_prod(A,A)),C4: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A5,B4) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,A5,C4) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,A5,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))),B4),C4)) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyR
tff(fact_4966_relcomp__UNION__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S2: set(product_prod(A,C)),R3: fun(D,set(product_prod(C,B))),I5: set(D)] : relcomp(A,C,B,S2,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),image2(D,set(product_prod(C,B)),R3),I5))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_xa(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S2),R3)),I5)) ).

% relcomp_UNION_distrib
tff(fact_4967_relcomp__UNION__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R3: fun(D,set(product_prod(A,C))),I5: set(D),S2: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(set(product_prod(A,C))),set(product_prod(A,C)),complete_Sup_Sup(set(product_prod(A,C))),aa(set(D),set(set(product_prod(A,C))),image2(D,set(product_prod(A,C)),R3),I5)),S2) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_xb(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R3),S2)),I5)) ).

% relcomp_UNION_distrib2
tff(fact_4968_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),X1: A,X2: A,Xs: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),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),X2),Xs))) = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs23)),Xs) ).

% mergesort_by_rel_split.simps(3)
tff(fact_4969_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_4970_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),X: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs12)),Xs23) ).

% mergesort_by_rel_split.simps(2)
tff(fact_4971_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A)),Xa2: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,X,Xa2) = Y )
     => ( ! [Xs1: list(A),Xs22: list(A)] :
            ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
           => ( ( Xa2 = 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)] :
              ( ( 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)),Xs1),Xs22) )
             => ! [X3: A] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),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),X3),Xs1)),Xs22) ) ) )
         => ~ ! [Xs1: list(A),Xs22: list(A)] :
                ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
               => ! [X12: A,X22: A,Xs2: list(A)] :
                    ( ( Xa2 = 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),X22),Xs2)) )
                   => ( 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),X22),Xs22)),Xs2) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_4972_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R2))
     => ( pp(aa(set(product_prod(B,C)),bool,finite_finite2(product_prod(B,C)),S))
       => ( relcomp(A,B,C,R2,S) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_xd(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S)),bot_bot(set(product_prod(A,C))),R2) ) ) ) ).

% relcomp_fold
tff(fact_4973_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),R2: set(product_prod(C,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S))
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert(product_prod(C,A)),X),R2),S) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_xe(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R2,S),S) ) ) ).

% insert_relcomp_fold
tff(fact_4974_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),X6: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S))
     => ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert(product_prod(C,A)),X),bot_bot(set(product_prod(C,A)))),S)),X6) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_xe(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X6,S) ) ) ).

% insert_relcomp_union_fold
tff(fact_4975_mergesort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),P: fun(fun(A,fun(A,bool)),fun(list(A),bool))] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),A0),A1)))
     => ( ! [R9: fun(A,fun(A,bool)),Xs2: list(A)] :
            ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R9),Xs2)))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
               => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R9),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)),Xs2)))) )
             => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
                 => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R9),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)),Xs2)))) )
               => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R9),Xs2)) ) ) )
       => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,A0),A1)) ) ) ).

% mergesort_by_rel.pinduct
tff(fact_4976_mergesort__by__rel__split_Opelims,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A)),Xa2: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),X),Xa2)))
       => ( ! [Xs1: list(A),Xs22: list(A)] :
              ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
             => ( ( Xa2 = 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) )
                 => ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,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)] :
                ( ( 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)),Xs1),Xs22) )
               => ! [X3: A] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),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),X3),Xs1)),Xs22) )
                     => ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,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),X3),nil(A))))) ) ) )
           => ~ ! [Xs1: list(A),Xs22: list(A)] :
                  ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
                 => ! [X12: A,X22: A,Xs2: list(A)] :
                      ( ( Xa2 = 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),X22),Xs2)) )
                     => ( ( 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),X22),Xs22)),Xs2) )
                       => ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,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),X22),Xs2))))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.pelims
tff(fact_4977_min__ext__compat,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),R2))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R2),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)))),insert(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,R2))) ) ).

% min_ext_compat
tff(fact_4978_max__ext__compat,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),R2))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R2),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)))),insert(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,R2))) ) ).

% max_ext_compat
tff(fact_4979_mergesort__by__rel_Opsimps,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R2),Xs)))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = Xs ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = merges9089515139780605204_merge(A,R2,aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)))) ) ) ) ) ).

% mergesort_by_rel.psimps
tff(fact_4980_mergesort__by__rel_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa2) = Y )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa2)))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
               => ( Y = Xa2 ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
               => ( Y = merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2)))) ) ) )
           => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa2))) ) ) ) ).

% mergesort_by_rel.pelims
tff(fact_4981_mergesort__by__rel__simps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] : aa(list(A),list(A),mergesort_by_rel(A,R2),nil(A)) = nil(A) ).

% mergesort_by_rel_simps(1)
tff(fact_4982_set__mergesort__by__rel,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R2),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_mergesort_by_rel
tff(fact_4983_mergesort__by__rel__merge__simps_I3_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Ys2: list(A)] : merges9089515139780605204_merge(A,R2,nil(A),Ys2) = Ys2 ).

% mergesort_by_rel_merge_simps(3)
tff(fact_4984_sorted__wrt__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y3))
          | pp(aa(A,bool,aa(A,fun(A,bool),R2,Y3),X3)) )
     => ( ! [X3: A,Y3: A,Z3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,Y3),Z3))
             => pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Z3)) ) )
       => ( sorted_wrt(A,R2,merges9089515139780605204_merge(A,R2,Xs,Ys2))
        <=> ( sorted_wrt(A,R2,Xs)
            & sorted_wrt(A,R2,Ys2) ) ) ) ) ).

% sorted_wrt_mergesort_by_rel_merge
tff(fact_4985_mergesort__by__rel__simps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A] : aa(list(A),list(A),mergesort_by_rel(A,R2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).

% mergesort_by_rel_simps(2)
tff(fact_4986_set__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] : aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R2,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ).

% set_mergesort_by_rel_merge
tff(fact_4987_mergesort__by__rel__simps_I3_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X1: A,X2: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R2),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),X2),Xs))) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aTP_Lamp_xf(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),R2)),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),X2),nil(A))),Xs)) ).

% mergesort_by_rel_simps(3)
tff(fact_4988_mergesort__by__rel__merge__simps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] : merges9089515139780605204_merge(A,R2,Xs,nil(A)) = Xs ).

% mergesort_by_rel_merge_simps(2)
tff(fact_4989_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys2: list(A)] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y))
       => ( merges9089515139780605204_merge(A,R2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,R2,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y))
       => ( merges9089515139780605204_merge(A,R2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2)) ) ) ) ).

% mergesort_by_rel_merge_simps(1)
tff(fact_4990_sorted__wrt__mergesort__by__rel,axiom,
    ! [X10: $tType,R2: fun(X10,fun(X10,bool)),Xs: list(X10)] :
      ( ! [X3: X10,Y3: X10] :
          ( pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,X3),Y3))
          | pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,Y3),X3)) )
     => ( ! [X3: X10,Y3: X10,Z3: X10] :
            ( pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,X3),Y3))
           => ( pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,Y3),Z3))
             => pp(aa(X10,bool,aa(X10,fun(X10,bool),R2,X3),Z3)) ) )
       => sorted_wrt(X10,R2,aa(list(X10),list(X10),mergesort_by_rel(X10,R2),Xs)) ) ) ).

% sorted_wrt_mergesort_by_rel
tff(fact_4991_sorted__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_by_rel(A,ord_less_eq(A)),Xs)) ) ).

% sorted_mergesort_by_rel
tff(fact_4992_mergesort__by__rel__merge_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,X,Xa2,Xb) = Y )
     => ( ! [X3: A,Xs2: list(A)] :
            ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
           => ! [Y3: A,Ys3: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
               => ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                     => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))) ) )
                    & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                     => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),merges9089515139780605204_merge(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys3)) ) ) ) ) )
       => ( ( ( Xb = nil(A) )
           => ( Y != Xa2 ) )
         => ~ ( ( Xa2 = 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_4993_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),V2: A,Va2: list(A)] : merges9089515139780605204_merge(A,R2,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_4994_sort__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_rm(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% sort_mergesort_by_rel
tff(fact_4995_max__ext__additive,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),R2: set(product_prod(A,A)),C4: set(A),D4: set(A)] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A5),B4)),max_ext(A,R2)))
     => ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),C4),D4)),max_ext(A,R2)))
       => pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),D4))),max_ext(A,R2))) ) ) ).

% max_ext_additive
tff(fact_4996_mergesort__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% mergesort_def
tff(fact_4997_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R2)))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
         => ( pp(aa(set(A),bool,finite_finite2(A),A22))
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),A1))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),A22))
                        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa4)),R2)) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_4998_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R2)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
        & pp(aa(set(A),bool,finite_finite2(A),A22))
        & ( A22 != bot_bot(set(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A1))
           => ? [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),A22))
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R2)) ) ) ) ) ).

% max_ext.simps
tff(fact_4999_max__ext_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
       => ( ( Y6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),Y6))
                    & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R2)) ) )
           => pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X6),Y6)),max_ext(A,R2))) ) ) ) ) ).

% max_ext.max_extI
tff(fact_5000_mergesort__by__rel_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa2) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( Y = Xa2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( Y = merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2)))) ) ) ) ) ).

% mergesort_by_rel.elims
tff(fact_5001_mergesort__by__rel_Osimps,axiom,
    ! [A: $tType,Xs: list(A),R2: fun(A,fun(A,bool))] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
       => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = Xs ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
       => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = merges9089515139780605204_merge(A,R2,aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)))) ) ) ) ).

% mergesort_by_rel.simps
tff(fact_5002_mergesort__by__rel__merge_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,X,Xa2,Xb) = Y )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa2),Xb))))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Y3: A,Ys3: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                 => ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                       => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))) ) )
                      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                       => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),merges9089515139780605204_merge(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys3)) ) ) )
                   => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))))) ) ) )
         => ( ( ( Xb = nil(A) )
             => ( ( Y = Xa2 )
               => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa2),nil(A))))) ) )
           => ~ ( ( Xa2 = 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) )
                     => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va))))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
tff(fact_5003_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S))
     => finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_xh(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_5004_max__ext__def,axiom,
    ! [A: $tType,X5: set(product_prod(A,A))] : max_ext(A,X5) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),max_extp(A,aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),X5)))) ).

% max_ext_def
tff(fact_5005_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_xj(fun(B,B),fun(A,fun(B,B)),F3)) ).

% comp_fun_commute_const
tff(fact_5006_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P: fun(A,bool)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_su(fun(A,bool),fun(A,fun(set(A),set(A))),P)) ).

% comp_fun_commute_filter_fold
tff(fact_5007_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_uk(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_5008_comp__fun__commute_Ofoldl__f__commute,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),A3: A,B2: B,Xs: list(A)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(B,B,aa(A,fun(B,B),F3,A3),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_xk(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),B2),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_xk(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),aa(B,B,aa(A,fun(B,B),F3,A3),B2)),Xs) ) ) ).

% comp_fun_commute.foldl_f_commute
tff(fact_5009_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_xl(set(A),fun(A,fun(B,fun(set(B),set(B)))),S))) ).

% comp_fun_commute_Image_fold
tff(fact_5010_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_xm(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B4)) ) ).

% comp_fun_commute_product_fold
tff(fact_5011_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X5: set(A),Xa3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)),X5),Xa3))
    <=> pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Xa3)),max_ext(A,R2))) ) ).

% max_extp_max_ext_eq
tff(fact_5012_max__extp__eq,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),X: set(A),Y: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R3),X),Y))
    <=> pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Y)),max_ext(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% max_extp_eq
tff(fact_5013_map__of__distinct__upd,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( map_add(A,B,fun_upd(A,option(B),aTP_Lamp_bk(A,option(B)),X,aa(B,option(B),some(B),Y)),map_of(A,B,Xs)) = fun_upd(A,option(B),map_of(A,B,Xs),X,aa(B,option(B),some(B),Y)) ) ) ).

% map_of_distinct_upd
tff(fact_5014_relpow__finite__bounded1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),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_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_xo(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_5015_range__prod,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),F3),top_top(set(C)))),product_Sigma(A,B,aa(set(C),set(A),image2(C,A,aa(fun(C,product_prod(A,B)),fun(C,A),comp(product_prod(A,B),A,C,product_fst(A,B)),F3)),top_top(set(C))),aTP_Lamp_xp(fun(C,product_prod(A,B)),fun(A,set(B)),F3)))) ).

% range_prod
tff(fact_5016_SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),B2: B,B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B4,A3)))
       => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A5,B4))) ) ) ).

% SigmaI
tff(fact_5017_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A5,B4)))
    <=> ( pp(aa(set(A),bool,member(A,A3),A5))
        & pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B4,A3))) ) ) ).

% mem_Sigma_iff
tff(fact_5018_map__add__find__right,axiom,
    ! [B: $tType,A: $tType,N2: fun(B,option(A)),K: B,Xx: A,M: fun(B,option(A))] :
      ( ( aa(B,option(A),N2,K) = aa(A,option(A),some(A),Xx) )
     => ( aa(B,option(A),map_add(B,A,M,N2),K) = aa(A,option(A),some(A),Xx) ) ) ).

% map_add_find_right
tff(fact_5019_Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),Q: fun(B,bool)] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_xq(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),P),Q))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P),aTP_Lamp_xr(fun(B,bool),fun(A,set(B)),Q)) ).

% Collect_case_prod
tff(fact_5020_empty__eq__map__add__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( ( aTP_Lamp_bk(A,option(B)) = map_add(A,B,F3,G3) )
    <=> ( ! [X4: A] : aa(A,option(B),F3,X4) = none(B)
        & ! [X4: A] : aa(A,option(B),G3,X4) = none(B) ) ) ).

% empty_eq_map_add_iff
tff(fact_5021_map__add__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,M,aTP_Lamp_bk(A,option(B))) = M ).

% map_add_empty
tff(fact_5022_empty__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,aTP_Lamp_bk(A,option(B)),M) = M ).

% empty_map_add
tff(fact_5023_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B4: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B4) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_5024_map__add__upd,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),G3: fun(A,option(B)),X: A,Y: B] : map_add(A,B,F3,fun_upd(A,option(B),G3,X,aa(B,option(B),some(B),Y))) = fun_upd(A,option(B),map_add(A,B,F3,G3),X,aa(B,option(B),some(B),Y)) ).

% map_add_upd
tff(fact_5025_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : product_Sigma(A,B,A5,aTP_Lamp_xs(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_5026_Times__empty,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( B4 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_5027_Sigma__UNIV__cancel,axiom,
    ! [B: $tType,A: $tType,A5: set(A),X6: 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,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),X6))),product_Sigma(A,B,A5,aTP_Lamp_xu(A,set(B)))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_UNIV_cancel
tff(fact_5028_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_xu(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A5),aTP_Lamp_xu(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_5029_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A5: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xt(set(B),fun(A,set(B)),A5))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xv(set(B),fun(A,set(B)),A5)) ).

% Compl_Times_UNIV1
tff(fact_5030_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [A6: A] :
            ( pp(aa(set(A),bool,member(A,A6),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,A6))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,B4))) ) ) ).

% finite_SigmaI
tff(fact_5031_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xu(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_5032_pairself__image__cart,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B4: 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,F3)),product_Sigma(B,B,A5,aTP_Lamp_xw(set(B),fun(B,set(B)),B4))) = product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F3),A5),aa(set(B),fun(A,set(A)),aTP_Lamp_xx(fun(B,A),fun(set(B),fun(A,set(A))),F3),B4)) ).

% pairself_image_cart
tff(fact_5033_finite__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N2: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
       => pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2))) ) ) ).

% finite_relpow
tff(fact_5034_fst__image__times,axiom,
    ! [B: $tType,A: $tType,B4: set(B),A5: set(A)] :
      ( ( ( B4 = bot_bot(set(B)) )
       => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))) = bot_bot(set(A)) ) )
      & ( ( B4 != bot_bot(set(B)) )
       => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))) = A5 ) ) ) ).

% fst_image_times
tff(fact_5035_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A5: set(B),B4: set(A)] :
      ( ( ( A5 = bot_bot(set(B)) )
       => ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A5,aTP_Lamp_mi(set(A),fun(B,set(A)),B4))) = bot_bot(set(A)) ) )
      & ( ( A5 != bot_bot(set(B)) )
       => ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A5,aTP_Lamp_mi(set(A),fun(B,set(A)),B4))) = B4 ) ) ) ).

% snd_image_times
tff(fact_5036_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys2)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_xy(list(B),fun(A,set(B)),Ys2)) ).

% set_product
tff(fact_5037_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A3: A,A5: set(A),B2: B,B4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5),aa(set(B),fun(A,set(B)),aTP_Lamp_xz(B,fun(set(B),fun(A,set(B))),B2),B4)) = 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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A5,aa(set(B),fun(A,set(B)),aTP_Lamp_xz(B,fun(set(B),fun(A,set(B))),B2),B4))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5),aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))) ).

% insert_Times_insert
tff(fact_5038_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,X3))) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A5,B4)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_me(fun(A,set(B)),fun(A,nat),B4)),A5) ) ) ) ).

% card_SigmaI
tff(fact_5039_Collect__case__prod__Sigma,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),Q: fun(A,fun(B,bool))] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ya(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P),aTP_Lamp_yb(fun(A,fun(B,bool)),fun(A,set(B)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_5040_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),C4: set(A),D4: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)) = product_Sigma(A,B,C4,aTP_Lamp_xt(set(B),fun(A,set(B)),D4)) )
    <=> ( ( ( A5 = C4 )
          & ( B4 = D4 ) )
        | ( ( ( A5 = bot_bot(set(A)) )
            | ( B4 = bot_bot(set(B)) ) )
          & ( ( C4 = bot_bot(set(A)) )
            | ( D4 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_5041_map__add__first__le,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M: fun(A,option(B)),M8: fun(A,option(B)),N2: fun(A,option(B))] :
          ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M8))
         => pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N2)),map_add(A,B,M8,N2))) ) ) ).

% map_add_first_le
tff(fact_5042_map__add__left__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),K: B,G3: fun(B,option(A))] :
      ( ( aa(B,option(A),F3,K) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F3,G3),K) = aa(B,option(A),G3,K) ) ) ).

% map_add_left_None
tff(fact_5043_map__add__find__left,axiom,
    ! [A: $tType,B: $tType,G3: fun(B,option(A)),K: B,F3: fun(B,option(A))] :
      ( ( aa(B,option(A),G3,K) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F3,G3),K) = aa(B,option(A),F3,K) ) ) ).

% map_add_find_left
tff(fact_5044_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),B4: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_yc(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,A5)),product_Sigma(A,B,I5,B4)) ).

% Sigma_Diff_distrib2
tff(fact_5045_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(A),C4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4),aTP_Lamp_xt(set(B),fun(A,set(B)),C4)) = 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,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))) ).

% Times_Diff_distrib1
tff(fact_5046_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J5: set(A),C4: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),J5),C4) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,C4)),product_Sigma(A,B,J5,C4)) ).

% Sigma_Diff_distrib1
tff(fact_5047_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C4: set(A),A5: set(B),B4: set(B)] :
      ( pp(aa(set(A),bool,member(A,X),C4))
     => ( ( product_Sigma(B,A,A5,aTP_Lamp_mi(set(A),fun(B,set(A)),C4)) = product_Sigma(B,A,B4,aTP_Lamp_mi(set(A),fun(B,set(A)),C4)) )
      <=> ( A5 = B4 ) ) ) ).

% Times_eq_cancel2
tff(fact_5048_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),C4: fun(A,set(B)),D4: fun(A,set(B))] :
      ( ( A5 = B4 )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),B4))
           => ( aa(A,set(B),C4,X3) = aa(A,set(B),D4,X3) ) )
       => ( product_Sigma(A,B,A5,C4) = product_Sigma(A,B,B4,D4) ) ) ) ).

% Sigma_cong
tff(fact_5049_relpow__Suc__D2_H,axiom,
    ! [A: $tType,N2: nat,R2: set(product_prod(A,A)),X5: A,Y4: A,Z5: A] :
      ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2)))
        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z5)),R2)) )
     => ? [W: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),W)),R2))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z5)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2))) ) ) ).

% relpow_Suc_D2'
tff(fact_5050_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A5,B4)))
     => ~ ( pp(aa(set(A),bool,member(A,A3),A5))
         => ~ pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B4,A3))) ) ) ).

% SigmaE2
tff(fact_5051_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A5,B4)))
     => pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B4,A3))) ) ).

% SigmaD2
tff(fact_5052_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A5,B4)))
     => pp(aa(set(A),bool,member(A,A3),A5)) ) ).

% SigmaD1
tff(fact_5053_SigmaE,axiom,
    ! [A: $tType,B: $tType,C2: product_prod(A,B),A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),C2),product_Sigma(A,B,A5,B4)))
     => ~ ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => ! [Y3: B] :
                ( pp(aa(set(B),bool,member(B,Y3),aa(A,set(B),B4,X3)))
               => ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) ) ) ) ) ).

% SigmaE
tff(fact_5054_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C4: set(A),B4: fun(A,set(B)),D4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,X3)),aa(A,set(B),D4,X3))) )
       => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A5,B4)),product_Sigma(A,B,C4,D4))) ) ) ).

% Sigma_mono
tff(fact_5055_Sigma__Int__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J5: set(A),C4: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I5),J5),C4) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I5,C4)),product_Sigma(A,B,J5,C4)) ).

% Sigma_Int_distrib1
tff(fact_5056_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J5: set(A),C4: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I5),J5),C4) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I5,C4)),product_Sigma(A,B,J5,C4)) ).

% Sigma_Un_distrib1
tff(fact_5057_member__product,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_product(A,B,A5,B4)))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))) ) ).

% member_product
tff(fact_5058_Product__Type_Oproduct__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] : product_product(A,B,A5,B4) = product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)) ).

% Product_Type.product_def
tff(fact_5059_map__add__SomeD,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N2: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N2),K) = aa(A,option(A),some(A),X) )
     => ( ( aa(B,option(A),N2,K) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N2,K) = none(A) )
          & ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_SomeD
tff(fact_5060_map__add__Some__iff,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N2: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N2),K) = aa(A,option(A),some(A),X) )
    <=> ( ( aa(B,option(A),N2,K) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N2,K) = none(A) )
          & ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_Some_iff
tff(fact_5061_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_rm(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_5062_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2)))
     => ( X = Y ) ) ).

% relpow_0_E
tff(fact_5063_relpow__0__I,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2))) ).

% relpow_0_I
tff(fact_5064_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z2: A,N2: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N2)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2)))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),R2)) ) ) ).

% relpow_Suc_E
tff(fact_5065_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y: A,N2: nat,R2: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N2)),R2))) ) ) ).

% relpow_Suc_I
tff(fact_5066_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z2: A,N2: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N2)),R2)))
     => ? [Y3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2))) ) ) ).

% relpow_Suc_D2
tff(fact_5067_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z2: A,N2: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N2)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2))) ) ) ).

% relpow_Suc_E2
tff(fact_5068_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A,N2: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N2)),R2))) ) ) ).

% relpow_Suc_I2
tff(fact_5069_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C4: set(A),A5: set(B),B4: set(B)] :
      ( pp(aa(set(A),bool,member(A,X),C4))
     => ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A5,aTP_Lamp_mi(set(A),fun(B,set(A)),C4))),product_Sigma(B,A,B4,aTP_Lamp_mi(set(A),fun(B,set(A)),C4))))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4)) ) ) ).

% Times_subset_cancel2
tff(fact_5070_mem__Times__iff,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
    <=> ( pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),X)),A5))
        & pp(aa(set(B),bool,member(B,aa(product_prod(A,B),B,product_snd(A,B),X)),B4)) ) ) ).

% mem_Times_iff
tff(fact_5071_in__prod__fst__sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A5: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),X)),A5))
     => ( pp(aa(set(B),bool,member(B,aa(product_prod(A,B),B,product_snd(A,B),X)),B4))
       => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))) ) ) ).

% in_prod_fst_sndI
tff(fact_5072_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B4)) ).

% card_cartesian_product
tff(fact_5073_swap__product,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B4: 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_wk(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A5,aTP_Lamp_mi(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),A5)) ).

% swap_product
tff(fact_5074_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I5: set(A),X6: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I5,X6) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),I5))
         => ( aa(A,set(B),X6,X4) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_5075_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(A),C4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4),aTP_Lamp_xt(set(B),fun(A,set(B)),C4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))) ).

% Times_Int_distrib1
tff(fact_5076_Sigma__Int__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),B4: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_yd(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I5,A5)),product_Sigma(A,B,I5,B4)) ).

% Sigma_Int_distrib2
tff(fact_5077_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),C4: set(A),D4: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))),product_Sigma(A,B,C4,aTP_Lamp_xt(set(B),fun(A,set(B)),D4))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4),aa(set(B),fun(A,set(B)),aTP_Lamp_ye(set(B),fun(set(B),fun(A,set(B))),B4),D4)) ).

% Times_Int_Times
tff(fact_5078_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))) ) ) ).

% finite_cartesian_product
tff(fact_5079_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(A),C4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4),aTP_Lamp_xt(set(B),fun(A,set(B)),C4)) = 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,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))) ).

% Times_Un_distrib1
tff(fact_5080_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),B4: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_yf(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I5,A5)),product_Sigma(A,B,I5,B4)) ).

% Sigma_Un_distrib2
tff(fact_5081_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,B)),A5: set(A),B4: set(B),S2: set(product_prod(B,C)),C4: set(C)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),bool),ord_less_eq(set(product_prod(B,C))),S2),product_Sigma(B,C,B4,aTP_Lamp_yg(set(C),fun(B,set(C)),C4))))
       => pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R3,S2)),product_Sigma(A,C,A5,aTP_Lamp_yh(set(C),fun(A,set(C)),C4)))) ) ) ).

% relcomp_subset_Sigma
tff(fact_5082_Sigma__Union,axiom,
    ! [B: $tType,A: $tType,X6: set(set(A)),B4: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X6),B4) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),image2(set(A),set(product_prod(A,B)),aTP_Lamp_yi(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B4)),X6)) ).

% Sigma_Union
tff(fact_5083_Id__on__subset__Times,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id_on(A,A5)),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ).

% Id_on_subset_Times
tff(fact_5084_map__add__def,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X5: A] : aa(A,option(B),map_add(A,B,M1,M22),X5) = case_option(option(B),B,aa(A,option(B),M1,X5),some(B),aa(A,option(B),M22,X5)) ).

% map_add_def
tff(fact_5085_relpowp__relpow__eq,axiom,
    ! [A: $tType,N2: nat,R2: set(product_prod(A,A)),X5: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N2),aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2))) ) ).

% relpowp_relpow_eq
tff(fact_5086_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,A5: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aTP_Lamp_xt(set(B),fun(A,set(B)),A5))) = aa(set(B),nat,finite_card(B),A5) ).

% card_cartesian_product_singleton
tff(fact_5087_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C4: set(B),B4: set(A),D4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),D4))))
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( C4 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),D4)) ) ) ) ).

% times_subset_iff
tff(fact_5088_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A5: set(C),B4: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_yk(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),product_Sigma(C,D,A5,aTP_Lamp_yl(set(D),fun(C,set(D)),B4))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F3),A5),aa(set(D),fun(A,set(B)),aTP_Lamp_ym(fun(D,B),fun(set(D),fun(A,set(B))),G3),B4)) ).

% image_paired_Times
tff(fact_5089_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_yn(set(A),fun(fun(A,set(B)),fun(A,bool)),A5),B4))))
     => ( ! [A6: A] :
            ( pp(aa(set(A),bool,member(A,A6),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,A6))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,B4))) ) ) ).

% finite_SigmaI2
tff(fact_5090_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
     => ( ( B4 != bot_bot(set(B)) )
       => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% finite_cartesian_productD1
tff(fact_5091_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
     => ( ( A5 != bot_bot(set(A)) )
       => pp(aa(set(B),bool,finite_finite2(B),B4)) ) ) ).

% finite_cartesian_productD2
tff(fact_5092_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( B4 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,finite_finite2(A),A5))
          & pp(aa(set(B),bool,finite_finite2(B),B4)) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_5093_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: 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,A5,B4)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_yn(set(A),fun(fun(A,set(B)),fun(A,bool)),A5),B4)) ).

% fst_image_Sigma
tff(fact_5094_relpow__E2,axiom,
    ! [A: $tType,X: A,Z2: A,N2: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2)))
     => ( ( ( N2 = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y3: A,M5: nat] :
              ( ( N2 = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R2))) ) ) ) ) ).

% relpow_E2
tff(fact_5095_relpow__E,axiom,
    ! [A: $tType,X: A,Z2: A,N2: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2)))
     => ( ( ( N2 = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y3: A,M5: nat] :
              ( ( N2 = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),R2)) ) ) ) ) ).

% relpow_E
tff(fact_5096_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E5: fun(C,set(A)),F4: fun(D,set(B)),A5: set(C),B4: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),image2(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_yp(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E5),F4))),product_Sigma(C,D,A5,aTP_Lamp_yl(set(D),fun(C,set(D)),B4)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),E5),A5)),aa(set(D),fun(A,set(B)),aTP_Lamp_yq(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B4)) ).

% UN_Times_distrib
tff(fact_5097_relpow__empty,axiom,
    ! [A: $tType,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_5098_sum_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,fun(C,A)),B4: set(C),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aTP_Lamp_dr(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B4)),A5) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7311177749621191930dd_sum(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3)),product_Sigma(B,C,A5,aTP_Lamp_yg(set(C),fun(B,set(C)),B4))) ) ).

% sum.cartesian_product
tff(fact_5099_prod_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,fun(C,A)),B4: set(C),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_ej(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B4)),A5) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7121269368397514597t_prod(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3)),product_Sigma(B,C,A5,aTP_Lamp_yg(set(C),fun(B,set(C)),B4))) ) ).

% prod.cartesian_product
tff(fact_5100_snd__image__Sigma,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B4: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A5,B4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5)) ).

% snd_image_Sigma
tff(fact_5101_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A5),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A5),aTP_Lamp_yr(set(product_prod(A,B)),fun(A,set(B)),A5)))) ).

% subset_fst_snd
tff(fact_5102_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B2: A,N2: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R2)))
    <=> ? [F10: fun(nat,A)] :
          ( ( aa(nat,A,F10,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F10,N2) = B2 )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N2))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F10,I4)),aa(nat,A,F10,aa(nat,nat,suc,I4)))),R2)) ) ) ) ).

% relpow_fun_conv
tff(fact_5103_sum_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: fun(B,set(C)),G3: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B4,X3))) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_ys(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B4),G3)),A5) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7311177749621191930dd_sum(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3)),product_Sigma(B,C,A5,B4)) ) ) ) ) ).

% sum.Sigma
tff(fact_5104_prod_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: fun(B,set(C)),G3: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B4,X3))) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_yt(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B4),G3)),A5) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7121269368397514597t_prod(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3)),product_Sigma(B,C,A5,B4)) ) ) ) ) ).

% prod.Sigma
tff(fact_5105_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_sy(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_5106_relpow__finite__bounded,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),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_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_yu(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ).

% relpow_finite_bounded
tff(fact_5107_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: fun(A,set(B))] : product_Sigma(A,B,A5,B4) = 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_yw(fun(A,set(B)),fun(A,set(product_prod(A,B))),B4)),A5)) ).

% Sigma_def
tff(fact_5108_product__fold,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => ( product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_yy(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B4),bot_bot(set(product_prod(A,B))),A5) ) ) ) ).

% product_fold
tff(fact_5109_lists__length__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),N2: nat] : aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_yz(set(A),fun(nat,fun(list(A),bool)),A5),N2)) = 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_uq(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_ln(set(A),fun(nat,fun(list(A),bool)),A5),N2)),aTP_Lamp_za(set(A),fun(list(A),set(A)),A5))) ).

% lists_length_Suc_eq
tff(fact_5110_ntrancl__def,axiom,
    ! [A: $tType,N2: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,N2,R2) = 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_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_zb(nat,fun(nat,bool),N2)))) ).

% ntrancl_def
tff(fact_5111_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),B4))
       => ~ pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))) ) ) ).

% infinite_cartesian_product
tff(fact_5112_Restr__subset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4)))),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))) ) ) ).

% Restr_subset
tff(fact_5113_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R3: set(product_prod(A,A)),As9: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R3,As9)
        <=> ! [I4: A,J3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I4),J3)),R3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As9,I4)),aa(A,B,As9,J3))) ) ) ) ).

% relChain_def
tff(fact_5114_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat))) ).

% natLess_def
tff(fact_5115_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( transitive_trancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_xo(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_5116_trancl__empty,axiom,
    ! [A: $tType] : transitive_trancl(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ).

% trancl_empty
tff(fact_5117_same__fst__trancl,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),R2: fun(A,set(product_prod(B,B)))] : transitive_trancl(product_prod(A,B),same_fst(A,B,P,R2)) = same_fst(A,B,P,aTP_Lamp_zc(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R2)) ).

% same_fst_trancl
tff(fact_5118_trancl__single,axiom,
    ! [A: $tType,A3: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A)))) ).

% trancl_single
tff(fact_5119_trancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V: set(product_prod(A,A)),X: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U2),V))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_trancl(A,U2)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_trancl(A,V))) ) ) ).

% trancl_mono_mp
tff(fact_5120_trancl__sub,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),transitive_trancl(A,R2))) ).

% trancl_sub
tff(fact_5121_trancl__mono,axiom,
    ! [A: $tType,P3: product_prod(A,A),R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_trancl(A,S2))) ) ) ).

% trancl_mono
tff(fact_5122_converse__trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2)),R3))
           => pp(aa(A,bool,P,Y3)) )
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_trancl(A,R3)))
               => ( pp(aa(A,bool,P,Z3))
                 => pp(aa(A,bool,P,Y3)) ) ) )
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% converse_trancl_induct
tff(fact_5123_trancl__trans__induct,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),P: fun(A,fun(A,bool))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R3)))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y3)) )
       => ( ! [X3: A,Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),transitive_trancl(A,R3)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y3))
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),transitive_trancl(A,R3)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y3),Z3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Z3)) ) ) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y)) ) ) ) ).

% trancl_trans_induct
tff(fact_5124_trancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_trancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R3))) ) ) ).

% trancl_into_trancl2
tff(fact_5125_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R3))) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5126_irrefl__trancl__rD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5127_converse__tranclE,axiom,
    ! [A: $tType,X: A,Z2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R3)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R3))
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R3))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),transitive_trancl(A,R3))) ) ) ) ).

% converse_tranclE
tff(fact_5128_r__r__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R2))) ) ) ).

% r_r_into_trancl
tff(fact_5129_trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),R3))
           => pp(aa(A,bool,P,Y3)) )
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_trancl(A,R3)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
               => ( pp(aa(A,bool,P,Y3))
                 => pp(aa(A,bool,P,Z3)) ) ) )
         => pp(aa(A,bool,P,B2)) ) ) ) ).

% trancl_induct
tff(fact_5130_trancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_trancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R3))) ) ) ).

% trancl_trans
tff(fact_5131_tranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ~ ! [C3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C3),B2)),R3)) ) ) ) ).

% tranclE
tff(fact_5132_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3))) ) ).

% trancl.r_into_trancl
tff(fact_5133_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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,R3)))
    <=> ( ? [A8: A,B13: A] :
            ( ( A1 = A8 )
            & ( A22 = B13 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A8),B13)),R3)) )
        | ? [A8: A,B13: A,C5: A] :
            ( ( A1 = A8 )
            & ( A22 = C5 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A8),B13)),transitive_trancl(A,R3)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),C5)),R3)) ) ) ) ).

% trancl.simps
tff(fact_5134_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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,R3)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,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)),R3))
       => ~ ! [B5: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B5)),transitive_trancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A22)),R3)) ) ) ) ).

% trancl.cases
tff(fact_5135_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_trancl(product_prod(A,B),R3)))
     => ( ! [A6: A,B5: B] :
            ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),R3))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,A6),B5)) )
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),transitive_trancl(product_prod(A,B),R3)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R3))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A6),B5))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_5136_trancl__sub__insert__trancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),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))),insert(product_prod(A,A)),X),R2)))) ).

% trancl_sub_insert_trancl
tff(fact_5137_trancl__unfold,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_trancl(A,R3) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),relcomp(A,A,A,transitive_trancl(A,R3),R3)) ).

% trancl_unfold
tff(fact_5138_trancl__subset__Sigma,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% trancl_subset_Sigma
tff(fact_5139_trancl__power,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_trancl(A,R2)))
    <=> ? [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% trancl_power
tff(fact_5140_trancl__Int__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_trancl(A,R3)),S2),R3)),S2))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),S2)) ) ) ).

% trancl_Int_subset
tff(fact_5141_Restr__trancl__mono,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),U2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,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_yj(set(A),fun(A,set(A)),U2))))))
     => pp(aa(set(product_prod(A,A)),bool,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_5142_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N2)),transitive_trancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ).

% less_eq
tff(fact_5143_trancl__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: 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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R3)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_zd(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R3)))) ).

% trancl_insert2
tff(fact_5144_nth__step__trancl,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),N2: nat,M: nat] :
      ( ! [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N5))),aa(nat,A,nth(A,Xs),N5))),R2)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N2)),aa(nat,A,nth(A,Xs),M))),transitive_trancl(A,R2))) ) ) ) ).

% nth_step_trancl
tff(fact_5145_map__to__set__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V2: B] : map_to_set(A,B,fun_upd(A,option(B),M,K,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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),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,M)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_ze(A,fun(product_prod(A,B),bool),K)))) ).

% map_to_set_upd
tff(fact_5146_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_yu(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_5147_rtrancl__last__visit__node,axiom,
    ! [A: $tType,S2: A,S4: A,R2: set(product_prod(A,A)),Sh: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S4)),transitive_rtrancl(A,R2)))
     => ( ( ( S2 != Sh )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S4)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_zf(A,fun(A,set(A)),Sh)))))) )
        | ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),Sh)),transitive_rtrancl(A,R2)))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S4)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_zf(A,fun(A,set(A)),Sh)))))) ) ) ) ).

% rtrancl_last_visit_node
tff(fact_5148_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_zg(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),P),Q))))
      <=> ! [Y5: A] :
            ( pp(aa(A,bool,P,Y5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),Y5)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5149_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: B,B4: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_zi(fun(B,A),fun(B,fun(set(B),fun(A,bool))),F3),A3),B4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F3,A3)),aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_zj(fun(B,A),fun(set(B),fun(A,bool)),F3),B4))) ).

% eq_or_mem_image_simp
tff(fact_5150_Eps__Opt__eq__None,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( eps_Opt(A,P) = none(A) )
    <=> ~ ? [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).

% Eps_Opt_eq_None
tff(fact_5151_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P: fun(B,fun(B,bool))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F3)),aa(fun(product_prod(B,B),bool),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,bool)),fun(product_prod(B,B),bool),product_case_prod(B,B,bool),P))) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_zk(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),F3),P)) ).

% pairself_image_eq
tff(fact_5152_converse__rtranclE_H,axiom,
    ! [A: $tType,U: A,V2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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_rtrancl(A,R2)))
     => ( ( U != V2 )
       => ~ ! [Vh: A] :
              ( ( U != Vh )
             => ( pp(aa(set(product_prod(A,A)),bool,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)),R2))
               => ~ pp(aa(set(product_prod(A,A)),bool,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)),transitive_rtrancl(A,R2))) ) ) ) ) ).

% converse_rtranclE'
tff(fact_5153_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_rtrancl(A,R3))) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_5154_converse__rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(A,bool,P,B2))
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_rtrancl(A,R3)))
               => ( pp(aa(A,bool,P,Z3))
                 => pp(aa(A,bool,P,Y3)) ) ) )
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% converse_rtrancl_induct
tff(fact_5155_converse__rtranclE,axiom,
    ! [A: $tType,X: A,Z2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R3)))
     => ( ( X != Z2 )
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R3))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),transitive_rtrancl(A,R3))) ) ) ) ).

% converse_rtranclE
tff(fact_5156_rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(A,bool,P,A3))
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_rtrancl(A,R3)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
               => ( pp(aa(A,bool,P,Y3))
                 => pp(aa(A,bool,P,Z3)) ) ) )
         => pp(aa(A,bool,P,B2)) ) ) ) ).

% rtrancl_induct
tff(fact_5157_rtrancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R3))) ) ) ).

% rtrancl_trans
tff(fact_5158_rtranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( ( A3 != B2 )
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_rtrancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2)),R3)) ) ) ) ).

% rtranclE
tff(fact_5159_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_rtrancl(A,R3))) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_5160_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A3: A,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),transitive_rtrancl(A,R3))) ).

% rtrancl.rtrancl_refl
tff(fact_5161_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R3)))
    <=> ( ? [A8: A] :
            ( ( A1 = A8 )
            & ( A22 = A8 ) )
        | ? [A8: A,B13: A,C5: A] :
            ( ( A1 = A8 )
            & ( A22 = C5 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A8),B13)),transitive_rtrancl(A,R3)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),C5)),R3)) ) ) ) ).

% rtrancl.simps
tff(fact_5162_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R3)))
     => ( ( A22 != A1 )
       => ~ ! [B5: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B5)),transitive_rtrancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A22)),R3)) ) ) ) ).

% rtrancl.cases
tff(fact_5163_Union__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),P: fun(B,bool)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(fun(B,bool),fun(set(A),bool),aTP_Lamp_zl(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),F3),P))) = aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_zm(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),F3),P)) ).

% Union_SetCompr_eq
tff(fact_5164_tranclp_Omono,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] : pp(aa(fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),bool,order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool))),aTP_Lamp_zn(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R3))) ).

% tranclp.mono
tff(fact_5165_rtranclp_Omono,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] : pp(aa(fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),bool,order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool))),aTP_Lamp_zo(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R3))) ).

% rtranclp.mono
tff(fact_5166_set__Cons__def,axiom,
    ! [A: $tType,A5: set(A),XS: set(list(A))] : set_Cons(A,A5,XS) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),bool),aTP_Lamp_zp(set(A),fun(set(list(A)),fun(list(A),bool)),A5),XS)) ).

% set_Cons_def
tff(fact_5167_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : pp(aa(fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),bool,order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),aTP_Lamp_zq(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less))) ).

% ord.lexordp.mono
tff(fact_5168_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => pp(aa(fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),bool,order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),aTP_Lamp_zr(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))))) ) ).

% lexordp.mono
tff(fact_5169_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aTP_Lamp_zs(fun(A,bool),fun(fun(A,B),fun(B,bool)),P),F3)))) ) ).

% finite_image_set
tff(fact_5170_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,bool),F3: fun(A,fun(B,C))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),Q)))
       => pp(aa(set(C),bool,finite_finite2(C),aa(fun(C,bool),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_zt(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),P),Q),F3)))) ) ) ).

% finite_image_set2
tff(fact_5171_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,transitive_rtrancl(A,R3))),transitive_rtrancl(list(A),listrel1(A,R3)))) ).

% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_5172_rtrancl__mono__rightI,axiom,
    ! [A: $tType,S: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),S3))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,S3))) ) ).

% rtrancl_mono_rightI
tff(fact_5173_rtrancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V: set(product_prod(A,A)),X: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U2),V))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,U2)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,V))) ) ) ).

% rtrancl_mono_mp
tff(fact_5174_r__le__rtrancl,axiom,
    ! [A: $tType,S: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,S))) ).

% r_le_rtrancl
tff(fact_5175_rtrancl__subset__rtrancl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),transitive_rtrancl(A,S2)))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),transitive_rtrancl(A,S2))) ) ).

% rtrancl_subset_rtrancl
tff(fact_5176_rtrancl__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,R2)))
       => ( transitive_rtrancl(A,S) = transitive_rtrancl(A,R2) ) ) ) ).

% rtrancl_subset
tff(fact_5177_rtrancl__mono,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),transitive_rtrancl(A,S2))) ) ).

% rtrancl_mono
tff(fact_5178_finite_Omono,axiom,
    ! [A: $tType] : pp(aa(fun(fun(set(A),bool),fun(set(A),bool)),bool,order_mono(fun(set(A),bool),fun(set(A),bool)),aTP_Lamp_zu(fun(set(A),bool),fun(set(A),bool)))) ).

% finite.mono
tff(fact_5179_in__rtrancl__UnI,axiom,
    ! [A: $tType,X: product_prod(A,A),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,R2)))
        | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,S))) )
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S)))) ) ).

% in_rtrancl_UnI
tff(fact_5180_rtrancl__Un__rtrancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),transitive_rtrancl(A,S))) = transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S)) ).

% rtrancl_Un_rtrancl
tff(fact_5181_in__rtrancl__insert,axiom,
    ! [A: $tType,X: product_prod(A,A),R2: set(product_prod(A,A)),R3: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,R2)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),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))),insert(product_prod(A,A)),R3),R2)))) ) ).

% in_rtrancl_insert
tff(fact_5182_fs__contract,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,fun(B,C)),S: set(C)] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_zv(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),F3),S))) = aa(fun(A,bool),set(A),collect(A),aa(set(C),fun(A,bool),aTP_Lamp_zw(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),F3),S)) ).

% fs_contract
tff(fact_5183_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P: fun(B,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_zx(fun(B,A),fun(fun(B,bool),fun(A,bool)),F3),P)) = aa(set(B),set(A),image2(B,A,F3),aa(fun(B,bool),set(B),collect(B),P)) ).

% setcompr_eq_image
tff(fact_5184_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_zj(fun(B,A),fun(set(B),fun(A,bool)),F3),A5)) = aa(set(B),set(A),image2(B,A,F3),A5) ).

% Setcompr_eq_image
tff(fact_5185_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Ys2: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),transitive_rtrancl(list(A),listrel1(A,R3))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),transitive_rtrancl(list(A),listrel1(A,R3)))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_5186_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R3))) ) ) ).

% trancl_rtrancl_trancl
tff(fact_5187_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_trancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R3))) ) ) ).

% rtrancl_trancl_trancl
tff(fact_5188_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R3))) ) ) ).

% rtrancl_into_trancl2
tff(fact_5189_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R3))) ) ) ).

% rtrancl_into_trancl1
tff(fact_5190_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)))
    <=> ( ( X = Y )
        | ( ( X != Y )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2))) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_5191_trancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3))) ) ).

% trancl_into_rtrancl
tff(fact_5192_tranclD2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ? [Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),transitive_rtrancl(A,R2)))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),R2)) ) ) ).

% tranclD2
tff(fact_5193_rtranclD,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2)))
     => ( ( A3 = B2 )
        | ( ( A3 != B2 )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))) ) ) ) ).

% rtranclD
tff(fact_5194_tranclD,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ? [Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),R2))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),transitive_rtrancl(A,R2))) ) ) ).

% tranclD
tff(fact_5195_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q))))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),transitive_rtrancl(A,P)))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),Q))
             => ( X3 = Y3 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P))) ) ) ).

% rtrancl_Un_separatorE
tff(fact_5196_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q))))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2)),transitive_rtrancl(A,P)))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),Q))
             => ( Y3 = X3 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P))) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_5197_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R3)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By))
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R3))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R3)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,A6),B5)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay)) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_5198_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xa2: A,Xb: B,Za: A,Zb: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R3)))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A6: A,B5: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),R3))
             => ~ pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R3))) ) ) ) ).

% converse_rtranclE2
tff(fact_5199_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R3)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay))
       => ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5))),transitive_rtrancl(product_prod(A,B),R3)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R3))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A6),B5))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).

% rtrancl_induct2
tff(fact_5200_rtrancl__Un__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),transitive_rtrancl(A,S))),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S)))) ).

% rtrancl_Un_subset
tff(fact_5201_rtrancl__sub__insert__rtrancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R2)),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))),insert(product_prod(A,A)),X),R2)))) ).

% rtrancl_sub_insert_rtrancl
tff(fact_5202_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_zy(fun(B,A),fun(A,bool),F3)) = aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_5203_finite__inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: A,A5: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),A5)) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_zz(A,fun(set(A),fun(A,bool)),A3),A5))) ) ).

% finite_inf_Sup
tff(fact_5204_ran__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : ran(A,B,M) = aa(fun(B,bool),set(B),collect(B),aTP_Lamp_aaa(fun(A,option(B)),fun(B,bool),M)) ).

% ran_def
tff(fact_5205_Pow__Compl,axiom,
    ! [A: $tType,A5: set(A)] : pow(A,aa(set(A),set(A),uminus_uminus(set(A)),A5)) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_aab(set(A),fun(set(A),bool),A5)) ).

% Pow_Compl
tff(fact_5206_listrel1__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : listrel1(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_aac(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% listrel1_def
tff(fact_5207_trancl__union__outside,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),U2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,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)))
       => ? [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),X3)),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))))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),U2))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),W2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E5),U2)))) ) ) ) ).

% trancl_union_outside
tff(fact_5208_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A)),X: A] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),transitive_rtrancl(list(A),listrel1(A,R3))))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))),transitive_rtrancl(list(A),listrel1(A,R3)))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_5209_trancl__over__edgeE,axiom,
    ! [A: $tType,U: A,W2: A,V1: A,V22: A,E5: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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))),insert(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))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,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)))
       => ~ ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V1)),transitive_rtrancl(A,E5)))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V22),W2)),transitive_rtrancl(A,E5))) ) ) ) ).

% trancl_over_edgeE
tff(fact_5210_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),transitive_rtrancl(list(A),listrel1(A,R3))))
     => ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_5211_Un__interval,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [B1: A,B22: A,B32: A,F3: fun(A,B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B1),B22))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_aad(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B22),F3))),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_aad(A,fun(A,fun(fun(A,B),fun(B,bool))),B22),B32),F3))) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_aad(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B32),F3)) ) ) ) ) ).

% Un_interval
tff(fact_5212_lex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_aae(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lex_conv
tff(fact_5213_Collect__ex__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aaf(fun(A,fun(B,bool)),fun(A,bool),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aah(fun(A,fun(B,bool)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_ex_eq
tff(fact_5214_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
       => ( ( A3 = B2 )
          | pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_5215_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: set(product_prod(A,C)),S2: set(product_prod(C,B))] : relcomp(A,C,B,R3,S2) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_aai(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R3),S2))) ).

% relcomp_unfold
tff(fact_5216_Restr__rtrancl__mono,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),U2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,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_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_yj(set(A),fun(A,set(A)),U2))))))
     => pp(aa(set(product_prod(A,A)),bool,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_rtrancl(A,E5))) ) ).

% Restr_rtrancl_mono
tff(fact_5217_graph__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_aaj(fun(A,option(B)),fun(product_prod(A,B),bool),M)) ).

% graph_def
tff(fact_5218_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N2)),transitive_rtrancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ).

% pred_nat_trancl_eq_le
tff(fact_5219_rtrancl__mapI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: A,E5: set(product_prod(A,A)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,E5)))
     => pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,A3)),aa(A,B,F3,B2))),transitive_rtrancl(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),image2(product_prod(A,A),product_prod(B,B),pairself(A,B,F3)),E5)))) ) ).

% rtrancl_mapI
tff(fact_5220_set__map__filter,axiom,
    ! [A: $tType,B: $tType,G3: fun(B,option(A)),Xs: list(B)] : aa(list(A),set(A),set2(A),map_filter(B,A,G3,Xs)) = aa(fun(A,bool),set(A),collect(A),aa(list(B),fun(A,bool),aTP_Lamp_aak(fun(B,option(A)),fun(list(B),fun(A,bool)),G3),Xs)) ).

% set_map_filter
tff(fact_5221_ex__assn__def,axiom,
    ! [A: $tType,P: fun(A,assn)] : ex_assn(A,P) = abs_assn(aTP_Lamp_aal(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),bool),P)) ).

% ex_assn_def
tff(fact_5222_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aam(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_5223_rtrancl__last__visit_H,axiom,
    ! [A: $tType,Q3: A,Q7: A,R2: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q7)),transitive_rtrancl(A,R2)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q7)),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))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_yj(set(A),fun(A,set(A)),S))))))
       => ~ ! [Qt: A] :
              ( pp(aa(set(A),bool,member(A,Qt),S))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt)),transitive_rtrancl(A,R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q7)),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))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_yj(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).

% rtrancl_last_visit'
tff(fact_5224_rtrancl__last__touch,axiom,
    ! [A: $tType,Q3: A,Q7: A,R2: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q7)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(A),bool,member(A,Q3),S))
       => ~ ! [Qt: A] :
              ( pp(aa(set(A),bool,member(A,Qt),S))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt)),transitive_rtrancl(A,R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q7)),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))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_yj(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).

% rtrancl_last_touch
tff(fact_5225_rtrancl__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R3: 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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R3)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aan(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R3)))) ).

% rtrancl_insert
tff(fact_5226_rtrancl__is__UN__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_rtrancl(A,R2) = 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_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),top_top(set(nat)))) ).

% rtrancl_is_UN_relpow
tff(fact_5227_rtrancl__imp__UN__relpow,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_rtrancl(A,R2)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),top_top(set(nat)))))) ) ).

% rtrancl_imp_UN_relpow
tff(fact_5228_rtrancl__last__visit,axiom,
    ! [A: $tType,Q3: A,Q7: A,R2: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q7)),transitive_rtrancl(A,R2)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q7)),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))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_yj(set(A),fun(A,set(A)),S))))))
       => ~ ! [Qt: A] :
              ( pp(aa(set(A),bool,member(A,Qt),S))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt)),transitive_trancl(A,R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q7)),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))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_yj(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).

% rtrancl_last_visit
tff(fact_5229_trancl__insert,axiom,
    ! [A: $tType,Y: A,X: A,R3: 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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R3)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R3)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aan(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Y),X),R3)))) ).

% trancl_insert
tff(fact_5230_set__drop__conv,axiom,
    ! [A: $tType,N2: nat,L: list(A)] : aa(list(A),set(A),set2(A),drop(A,N2,L)) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_aao(nat,fun(list(A),fun(A,bool)),N2),L)) ).

% set_drop_conv
tff(fact_5231_trancl__multi__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),X6: set(A),M: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,aa(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))),R3),product_Sigma(A,A,X6,aTP_Lamp_aap(A,fun(A,set(A)),M))))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
       => ~ ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),transitive_rtrancl(A,R3)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),B2)),transitive_rtrancl(A,R3))) ) ) ) ) ).

% trancl_multi_insert
tff(fact_5232_trancl__multi__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),M: A,X6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,aa(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))),R3),product_Sigma(A,A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),M),bot_bot(set(A))),aTP_Lamp_yj(set(A),fun(A,set(A)),X6))))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
       => ~ ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),M)),transitive_rtrancl(A,R3)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2)),transitive_rtrancl(A,R3))) ) ) ) ) ).

% trancl_multi_insert2
tff(fact_5233_brk__rel__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : brk_rel(A,B,R2) = aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B))),aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),fun(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B)))),sup_sup(set(product_prod(product_prod(bool,A),product_prod(bool,B)))),aa(fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),set(product_prod(product_prod(bool,A),product_prod(bool,B))),collect(product_prod(product_prod(bool,A),product_prod(bool,B))),aTP_Lamp_aaq(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),R2))),aa(fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),set(product_prod(product_prod(bool,A),product_prod(bool,B))),collect(product_prod(product_prod(bool,A),product_prod(bool,B))),aTP_Lamp_aar(product_prod(product_prod(bool,A),product_prod(bool,B)),bool))) ).

% brk_rel_def
tff(fact_5234_lexn__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),N2: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N2) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_aas(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R3),N2))) ).

% lexn_conv
tff(fact_5235_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: set(C),F3: fun(C,A),G3: fun(C,B)] : bNF_Greatest_image2(C,A,B,A5,F3,G3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_aat(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A5),F3),G3)) ).

% image2_def
tff(fact_5236_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F3: fun(D,B)] :
          ( pp(aa(fun(A,fun(B,C)),bool,order_mono(A,fun(B,C)),Q))
         => pp(aa(fun(A,fun(D,C)),bool,order_mono(A,fun(D,C)),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aau(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F3))) ) ) ).

% mono_compose
tff(fact_5237_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_5238_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F3: fun(B,A),X: B,C2: C,G3: fun(B,C),A5: set(B)] :
      ( ( B2 = aa(B,A,F3,X) )
     => ( ( C2 = aa(B,C,G3,X) )
       => ( pp(aa(set(B),bool,member(B,X),A5))
         => pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C2)),bNF_Greatest_image2(B,A,C,A5,F3,G3))) ) ) ) ).

% image2_eqI
tff(fact_5239_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A)),N2: nat] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N2)))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N2 )
        & ( aa(list(A),nat,size_size(list(A)),Ys2) = N2 ) ) ) ).

% lexn_length
tff(fact_5240_lex__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = 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,R3)),top_top(set(nat)))) ).

% lex_def
tff(fact_5241_lexord__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lexord(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_aav(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lexord_def
tff(fact_5242_list__collect__set__alt,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F3,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),bool),aTP_Lamp_aaw(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F3),L))) ).

% list_collect_set_alt
tff(fact_5243_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),N2: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),aa(nat,nat,suc,N2)) = 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),R3,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N2)))),aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_aax(nat,fun(list(A),fun(list(A),bool)),N2)))) ).

% lexn.simps(2)
tff(fact_5244_map__prod__ident,axiom,
    ! [B: $tType,A: $tType,X5: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_cq(A,A),aTP_Lamp_aay(B,B)),X5) = X5 ).

% map_prod_ident
tff(fact_5245_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,A3)),aa(D,B,G3,B2)) ).

% map_prod_simp
tff(fact_5246_fst__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),X: product_prod(C,D)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),X)) = aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),X)) ).

% fst_map_prod
tff(fact_5247_snd__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,B),G3: fun(D,A),X: product_prod(C,D)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,D),product_prod(B,A),product_map_prod(C,B,D,A,F3,G3),X)) = aa(D,A,G3,aa(product_prod(C,D),D,product_snd(C,D),X)) ).

% snd_map_prod
tff(fact_5248_list__collect__set__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),A3: B] : list_collect_set(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),nil(B))) = aa(B,set(A),F3,A3) ).

% list_collect_set_simps(2)
tff(fact_5249_list__collect__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,set(A))] : list_collect_set(B,A,F3,nil(B)) = bot_bot(set(A)) ).

% list_collect_set_simps(1)
tff(fact_5250_list__collect__set__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),A3: B,L: list(B)] : list_collect_set(B,A,F3,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),F3,A3)),list_collect_set(B,A,F3,L)) ).

% list_collect_set_simps(3)
tff(fact_5251_list__collect__set__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B),L3: list(B)] : list_collect_set(B,A,F3,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F3,L)),list_collect_set(B,A,F3,L3)) ).

% list_collect_set_simps(4)
tff(fact_5252_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),F3: fun(A,C),G3: fun(B,D)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2))
     => pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F3,A3)),aa(B,D,G3,B2))),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F3,G3)),R2))) ) ).

% map_prod_imageI
tff(fact_5253_lexord__cons__cons,axiom,
    ! [A: $tType,A3: A,X: list(A),B2: A,Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R3)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
        | ( ( A3 = B2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R3))) ) ) ) ).

% lexord_cons_cons
tff(fact_5254_lexord__Nil__left,axiom,
    ! [A: $tType,Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Y)),lexord(A,R3)))
    <=> ? [A8: A,X4: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A8),X4) ) ).

% lexord_Nil_left
tff(fact_5255_list__collect__set__map__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,set(A)),X: fun(C,B),A3: C] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),nil(C)))) = aa(B,set(A),F3,aa(C,B,X,A3)) ).

% list_collect_set_map_simps(2)
tff(fact_5256_list__collect__set__map__simps_I1_J,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,set(A)),X: fun(C,B)] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),nil(C))) = bot_bot(set(A)) ).

% list_collect_set_map_simps(1)
tff(fact_5257_list__collect__set__map__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,set(A)),X: fun(C,B),A3: C,L: list(C)] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),L))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F3,aa(C,B,X,A3))),list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),L))) ).

% list_collect_set_map_simps(3)
tff(fact_5258_list__collect__set__map__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,set(A)),X: fun(C,B),L: list(C),L3: list(C)] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L),L3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),L))),list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),L3))) ).

% list_collect_set_map_simps(4)
tff(fact_5259_fst__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,C),G3: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),comp(product_prod(C,D),C,product_prod(A,B),product_fst(C,D)),product_map_prod(A,C,B,D,F3,G3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_map_prod
tff(fact_5260_snd__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,D),G3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),comp(product_prod(D,C),C,product_prod(A,B),product_snd(D,C)),product_map_prod(A,D,B,C,F3,G3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),G3),product_snd(A,B)) ).

% snd_comp_map_prod
tff(fact_5261_map__prod_Ocomp,axiom,
    ! [A: $tType,C: $tType,E: $tType,F: $tType,D: $tType,B: $tType,F3: fun(C,E),G3: fun(D,F),H: fun(A,C),I2: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F)),comp(product_prod(C,D),product_prod(E,F),product_prod(A,B),product_map_prod(C,E,D,F,F3,G3)),product_map_prod(A,C,B,D,H,I2)) = product_map_prod(A,E,B,F,aa(fun(A,C),fun(A,E),comp(C,E,A,F3),H),aa(fun(B,D),fun(B,F),comp(D,F,B,G3),I2)) ).

% map_prod.comp
tff(fact_5262_map__prod_Ocompositionality,axiom,
    ! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: fun(C,E),G3: fun(D,F),H: fun(A,C),I2: fun(B,D),Prod: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F),product_map_prod(C,E,D,F,F3,G3),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,H,I2),Prod)) = aa(product_prod(A,B),product_prod(E,F),product_map_prod(A,E,B,F,aa(fun(A,C),fun(A,E),comp(C,E,A,F3),H),aa(fun(B,D),fun(B,F),comp(D,F,B,G3),I2)),Prod) ).

% map_prod.compositionality
tff(fact_5263_map__prod__compose,axiom,
    ! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F,D),G22: fun(B,F)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),comp(E,C,A,F1),F22),aa(fun(B,F),fun(B,D),comp(F,D,B,G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F)),fun(product_prod(A,B),product_prod(C,D)),comp(product_prod(E,F),product_prod(C,D),product_prod(A,B),product_map_prod(E,C,F,D,F1,G1)),product_map_prod(A,E,B,F,F22,G22)) ).

% map_prod_compose
tff(fact_5264_case__prod__map__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: fun(B,fun(C,A)),F3: fun(D,B),G3: fun(E,C),X: product_prod(D,E)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H),aa(product_prod(D,E),product_prod(B,C),product_map_prod(D,B,E,C,F3,G3),X)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_case_prod(D,E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aaz(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),H),F3),G3)),X) ).

% case_prod_map_prod
tff(fact_5265_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F3: fun(C,A),G3: fun(D,B),R2: set(product_prod(C,D))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),C2),aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3)),R2)))
     => ~ ! [X3: C,Y3: D] :
            ( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),aa(D,B,G3,Y3)) )
           => ~ pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X3),Y3)),R2)) ) ) ).

% prod_fun_imageE
tff(fact_5266_lexord__irreflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lexord(A,R3))) ) ).

% lexord_irreflexive
tff(fact_5267_lexord__linear,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: list(A),Y: list(A)] :
      ( ! [A6: A,B5: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B5)),R3))
          | ( A6 = B5 )
          | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6)),R3)) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R3)))
        | ( X = Y )
        | pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X)),lexord(A,R3))) ) ) ).

% lexord_linear
tff(fact_5268_lexord__Nil__right,axiom,
    ! [A: $tType,X: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))),lexord(A,R3))) ).

% lexord_Nil_right
tff(fact_5269_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V2: list(A),R3: set(product_prod(A,A)),X: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V2)),lexord(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V2))),lexord(A,R3))) ) ).

% lexord_append_leftI
tff(fact_5270_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(A,C),G3: fun(B,D)] : product_map_prod(A,C,B,D,F3,G3) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_aba(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F3),G3)) ).

% map_prod_def
tff(fact_5271_case__prod__o__map__prod,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F3: fun(D,fun(E,C)),G1: fun(A,D),G22: fun(B,E)] : aa(fun(product_prod(A,B),product_prod(D,E)),fun(product_prod(A,B),C),comp(product_prod(D,E),C,product_prod(A,B),aa(fun(D,fun(E,C)),fun(product_prod(D,E),C),product_case_prod(D,E,C),F3)),product_map_prod(A,D,B,E,G1,G22)) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_abb(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),F3),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_5272_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A)),Ys2: list(A),Zs: list(A)] :
      ( ! [X3: A,Y3: A,Z3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3)),R3)) ) ) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lexord(A,R3)))
       => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys2),Zs)),lexord(A,R3)))
         => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),lexord(A,R3))) ) ) ) ).

% lexord_partial_trans
tff(fact_5273_lexord__append__leftD,axiom,
    ! [A: $tType,X: list(A),U: list(A),V2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V2))),lexord(A,R3)))
     => ( ! [A6: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),A6)),R3))
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V2)),lexord(A,R3))) ) ) ).

% lexord_append_leftD
tff(fact_5274_lexord__append__rightI,axiom,
    ! [A: $tType,Y: list(A),X: list(A),R3: set(product_prod(A,A))] :
      ( ? [B11: A,Z5: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B11),Z5)
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),Y))),lexord(A,R3))) ) ).

% lexord_append_rightI
tff(fact_5275_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys2: list(A),Qs: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Qs))),lexord(A,R3)))
     => ( ( Xs != Ys2 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lexord(A,R3))) ) ) ) ) ).

% lexord_sufE
tff(fact_5276_list__collect__set__as__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F3,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)),map(B,set(A),F3),L))) ).

% list_collect_set_as_map
tff(fact_5277_map__prod__surj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(B,A),A5: set(B),A15: set(A),G3: fun(D,C),B4: set(D),B14: set(C)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A5) = A15 )
     => ( ( aa(set(D),set(C),image2(D,C,G3),B4) = B14 )
       => ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F3,G3)),product_Sigma(B,D,A5,aTP_Lamp_abc(set(D),fun(B,set(D)),B4))) = product_Sigma(A,C,A15,aTP_Lamp_yh(set(C),fun(A,set(C)),B14)) ) ) ) ).

% map_prod_surj_on
tff(fact_5278_lexord__lex,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lex(A,R3)))
    <=> ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R3)))
        & ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_5279_map__prod__surj,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(A,B),G3: fun(C,D)] :
      ( ( aa(set(A),set(B),image2(A,B,F3),top_top(set(A))) = top_top(set(B)) )
     => ( ( aa(set(C),set(D),image2(C,D,G3),top_top(set(C))) = top_top(set(D)) )
       => ( aa(set(product_prod(A,C)),set(product_prod(B,D)),image2(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F3,G3)),top_top(set(product_prod(A,C)))) = top_top(set(product_prod(B,D))) ) ) ) ).

% map_prod_surj
tff(fact_5280_lexord__append__left__rightI,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)))),lexord(A,R3))) ) ).

% lexord_append_left_rightI
tff(fact_5281_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R3)))
    <=> ( ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R3)) )
        | pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys2),Zs)),lexord(A,R3))) ) ) ).

% lexord_same_pref_iff
tff(fact_5282_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R3: set(product_prod(A,A)),V2: list(A),Z2: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2)),lexord(A,R3)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,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,R3))) ) ) ).

% lexord_sufI
tff(fact_5283_list__collect__set__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F3,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),bool),aTP_Lamp_abd(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F3),L))) ).

% list_collect_set_def
tff(fact_5284_List_Olexordp__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] :
      ( lexordp(A,R3,Xs,Ys2)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% List.lexordp_def
tff(fact_5285_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,B),B4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A5,F3)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B4)) ) ).

% Gr_incl
tff(fact_5286_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),X: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G3),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G3,aa(product_prod(C,D),D,product_snd(C,D),X))) ).

% apfst_apsnd
tff(fact_5287_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F3: fun(C,A),X: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X)),Y) ).

% apfst_conv
tff(fact_5288_apfst__eq__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),X: product_prod(C,B),G3: fun(C,A)] :
      ( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),X) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G3),X) )
    <=> ( aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),X)) = aa(C,A,G3,aa(product_prod(C,B),C,product_fst(C,B),X)) ) ) ).

% apfst_eq_conv
tff(fact_5289_fst__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),X: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),X)) = aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),X)) ).

% fst_apfst
tff(fact_5290_snd__apfst,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,B),X: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F3),X)) = aa(product_prod(C,A),A,product_snd(C,A),X) ).

% snd_apfst
tff(fact_5291_snd__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),comp(product_prod(C,B),B,product_prod(A,B),product_snd(C,B)),product_apfst(A,C,B,F3)) = product_snd(A,B) ).

% snd_comp_apfst
tff(fact_5292_fst__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),comp(product_prod(C,B),C,product_prod(A,B),product_fst(C,B)),product_apfst(A,C,B,F3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_apfst
tff(fact_5293_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G3,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F3,aa(product_prod(D,C),C,product_snd(D,C),X))) ).

% apsnd_apfst
tff(fact_5294_apsnd__apfst__commute,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),P3: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),P3)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G3),aa(product_prod(D,C),product_prod(D,B),aa(fun(C,B),fun(product_prod(D,C),product_prod(D,B)),product_apsnd(C,B,D),F3),P3)) ).

% apsnd_apfst_commute
tff(fact_5295_apfst__compose,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,C),X: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G3),X)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),comp(C,A,D,F3),G3)),X) ).

% apfst_compose
tff(fact_5296_GrD2,axiom,
    ! [A: $tType,B: $tType,X: A,Fx: B,A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A5,F3)))
     => ( aa(A,B,F3,X) = Fx ) ) ).

% GrD2
tff(fact_5297_GrD1,axiom,
    ! [B: $tType,A: $tType,X: A,Fx: B,A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A5,F3)))
     => pp(aa(set(A),bool,member(A,X),A5)) ) ).

% GrD1
tff(fact_5298_Gr__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] : bNF_Gr(A,B,A5,F3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_abe(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A5),F3)) ).

% Gr_def
tff(fact_5299_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q3: product_prod(A,B),F3: fun(C,A),P3: product_prod(C,B)] :
      ( ( Q3 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),P3) )
     => ~ ! [X3: C,Y3: B] :
            ( ( P3 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Y3) )
           => ( Q3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),Y3) ) ) ) ).

% apfst_convE
tff(fact_5300_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R3)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
          & ( take(A,aa(list(A),nat,size_size(list(A)),X),Y) = X ) )
        | ? [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))))
            & ( take(A,I4,X) = take(A,I4,Y) )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,X),I4)),aa(nat,A,nth(A,Y),I4))),R3)) ) ) ) ).

% lexord_take_index_conv
tff(fact_5301_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P3: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P3) )
    <=> ? [A8: B] : P3 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A8),B2) ) ).

% eq_snd_iff
tff(fact_5302_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_5303_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).

% min.left_idem
tff(fact_5304_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).

% min.right_idem
tff(fact_5305_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = M )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),N2)) ) ) ).

% min_eq_arg(1)
tff(fact_5306_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = N2 )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N2),M)) ) ) ).

% min_eq_arg(2)
tff(fact_5307_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N2: A,M: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N2),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2)))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = N2 ) ) ) ).

% min_arg_le(1)
tff(fact_5308_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,N2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2)))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = M ) ) ) ).

% min_arg_le(2)
tff(fact_5309_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% min.bounded_iff
tff(fact_5310_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5311_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb1
tff(fact_5312_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2)),M))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_5313_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2)),N2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = N2 ) ) ) ).

% min_arg_not_ge(2)
tff(fact_5314_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% min_less_self_conv(1)
tff(fact_5315_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% min_less_self_conv(2)
tff(fact_5316_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min_simps(1)
tff(fact_5317_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min_simps(2)
tff(fact_5318_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb3
tff(fact_5319_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_5320_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y)) ) ) ) ).

% min_less_iff_conj
tff(fact_5321_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).

% min_bot
tff(fact_5322_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_5323_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).

% min_top
tff(fact_5324_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).

% min_top2
tff(fact_5325_min__0R,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N2),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_5326_min__0L,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N2) = zero_zero(nat) ).

% min_0L
tff(fact_5327_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N2)) ).

% min_Suc_Suc
tff(fact_5328_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(1)
tff(fact_5329_Int__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) ) ).

% Int_atMost
tff(fact_5330_min__Suc__gt_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),aa(nat,nat,suc,A3)) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(2)
tff(fact_5331_min__Suc__gt_I1_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A3)),B2) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(1)
tff(fact_5332_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(4)
tff(fact_5333_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(3)
tff(fact_5334_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(2)
tff(fact_5335_Int__atLeastAtMostR1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),B2)),set_or1337092689740270186AtMost(A,C2,D3)) = set_or1337092689740270186AtMost(A,C2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMostR1
tff(fact_5336_Int__atLeastAtMostL1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(A,set(A),set_ord_atMost(A),D3)) = set_or1337092689740270186AtMost(A,A3,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMostL1
tff(fact_5337_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.strict_coboundedI2
tff(fact_5338_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.strict_coboundedI1
tff(fact_5339_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_5340_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).

% min.strict_boundedE
tff(fact_5341_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2)) ) ) ) ).

% min_less_iff_disj
tff(fact_5342_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_5343_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N2),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).

% nat_mult_min_right
tff(fact_5344_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N2)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)) ).

% nat_mult_min_left
tff(fact_5345_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.assoc
tff(fact_5346_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3) ) ).

% min.commute
tff(fact_5347_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B2),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.left_commute
tff(fact_5348_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa3) = X5 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa3) = Xa3 ) ) ) ) ).

% min_def_raw
tff(fact_5349_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% min_le_iff_disj
tff(fact_5350_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.coboundedI2
tff(fact_5351_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).

% min.coboundedI1
tff(fact_5352_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_5353_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_5354_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2)) ) ).

% min.cobounded2
tff(fact_5355_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3)) ) ).

% min.cobounded1
tff(fact_5356_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.order_iff
tff(fact_5357_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))) ) ) ) ).

% min.boundedI
tff(fact_5358_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).

% min.boundedE
tff(fact_5359_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% min.orderI
tff(fact_5360_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.orderE
tff(fact_5361_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D3))) ) ) ) ).

% min.mono
tff(fact_5362_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_5363_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).

% min_absorb1
tff(fact_5364_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ) ).

% min_def
tff(fact_5365_min__diff,axiom,
    ! [M: nat,I2: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),I2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),I2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N2)),I2) ).

% min_diff
tff(fact_5366_complete__linorder__inf__min,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ( inf_inf(A) = ord_min(A) ) ) ).

% complete_linorder_inf_min
tff(fact_5367_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5368_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_5369_min__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),M: A,N2: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F3,M)),aa(A,B,F3,N2)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2)) ) ) ) ).

% min_of_mono
tff(fact_5370_greaterThan__Int__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) ) ).

% greaterThan_Int_greaterThan
tff(fact_5371_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_5372_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( ( S = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = X ) )
            & ( ( S != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,complete_Inf_Inf(A),S)) ) ) ) ) ) ).

% Inf_insert_finite
tff(fact_5373_min__Suc2,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N2)) = case_nat(nat,zero_zero(nat),aTP_Lamp_abf(nat,fun(nat,nat),N2),M) ).

% min_Suc2
tff(fact_5374_min__Suc1,axiom,
    ! [N2: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N2)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_abg(nat,fun(nat,nat),N2),M) ).

% min_Suc1
tff(fact_5375_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,P3: product_prod(A,B)] :
      ( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P3) )
    <=> ? [B13: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B13) ) ).

% eq_fst_iff
tff(fact_5376_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),A,case_list(A,A,X,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_abh(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_5377_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_abi(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys2)) ).

% set_zip
tff(fact_5378_rtrancl__restrictI,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,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_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_yj(set(A),fun(A,set(A)),R2))))))
     => ( ~ pp(aa(set(A),bool,member(A,U),R2))
       => pp(aa(set(product_prod(A,A)),bool,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_rtrancl(A,rel_restrict(A,E5,R2)))) ) ) ).

% rtrancl_restrictI
tff(fact_5379_zip__eq__zip__same__len,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),A4: list(A),B3: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
       => ( ( zip(A,B,A3,B2) = zip(A,B,A4,B3) )
        <=> ( ( A3 = A4 )
            & ( B2 = B3 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5380_rel__restrict__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : rel_restrict(A,R2,bot_bot(set(A))) = R2 ).

% rel_restrict_empty
tff(fact_5381_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys2: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),zip(A,B,Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5382_nth__zip,axiom,
    ! [A: $tType,B: $tType,I2: nat,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys2)))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys2)),I2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5383_rel__restrict__sub,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R2,A5)),R2)) ).

% rel_restrict_sub
tff(fact_5384_rel__restrict__mono,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),B4: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A5),B4))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A5,R2)),rel_restrict(A,B4,R2))) ) ).

% rel_restrict_mono
tff(fact_5385_finite__rel__restrict,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),rel_restrict(A,R2,A5))) ) ).

% finite_rel_restrict
tff(fact_5386_zip__inj,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),A4: list(A),B3: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
       => ( ( zip(A,B,A3,B2) = zip(A,B,A4,B3) )
         => ( ( A3 = A4 )
            & ( B2 = B3 ) ) ) ) ) ).

% zip_inj
tff(fact_5387_pair__list__split,axiom,
    ! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
      ~ ! [L12: list(A),L23: list(B)] :
          ( ( L = zip(A,B,L12,L23) )
         => ( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L23) )
           => ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L23) ) ) ) ).

% pair_list_split
tff(fact_5388_rel__restrict__lift,axiom,
    ! [A: $tType,X: A,Y: A,E5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,E5,R2)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),E5)) ) ).

% rel_restrict_lift
tff(fact_5389_rel__restrictI,axiom,
    ! [A: $tType,X: A,R2: set(A),Y: A,E5: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,member(A,X),R2))
     => ( ~ pp(aa(set(A),bool,member(A,Y),R2))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),E5))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,E5,R2))) ) ) ) ).

% rel_restrictI
tff(fact_5390_rel__restrict__notR_I1_J,axiom,
    ! [A: $tType,X: A,Y: A,A5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,A5,R2)))
     => ~ pp(aa(set(A),bool,member(A,X),R2)) ) ).

% rel_restrict_notR(1)
tff(fact_5391_rel__restrict__notR_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,A5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,A5,R2)))
     => ~ pp(aa(set(A),bool,member(A,Y),R2)) ) ).

% rel_restrict_notR(2)
tff(fact_5392_rel__restrict__union,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A),B4: set(A)] : rel_restrict(A,R2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = rel_restrict(A,rel_restrict(A,R2,A5),B4) ).

% rel_restrict_union
tff(fact_5393_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys2: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys2,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_abj(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys2),Zs)) ).

% zip_assoc
tff(fact_5394_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,X: A,Ys2: list(B),Y: B] : zip(A,B,list_update(A,Xs,I2,X),list_update(B,Ys2,I2,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys2),I2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_update
tff(fact_5395_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys2: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys2,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_abl(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys2,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_5396_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_uv(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_5397_rel__restrict__trancl__notR_I2_J,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,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,R2))))
     => ~ pp(aa(set(A),bool,member(A,W2),R2)) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_5398_rel__restrict__trancl__notR_I1_J,axiom,
    ! [A: $tType,V2: A,W2: A,E5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,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,R2))))
     => ~ pp(aa(set(A),bool,member(A,V2),R2)) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_5399_rel__restrict__trancl__mem,axiom,
    ! [A: $tType,A3: A,B2: A,A5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,rel_restrict(A,A5,R2))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),rel_restrict(A,transitive_trancl(A,A5),R2))) ) ).

% rel_restrict_trancl_mem
tff(fact_5400_rel__restrict__mono2,axiom,
    ! [A: $tType,R2: set(A),S: set(A),A5: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R2),S))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A5,S)),rel_restrict(A,A5,R2))) ) ).

% rel_restrict_mono2
tff(fact_5401_rel__restrict__trancl__sub,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),R2: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,rel_restrict(A,A5,R2))),rel_restrict(A,transitive_trancl(A,A5),R2))) ).

% rel_restrict_trancl_sub
tff(fact_5402_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),zip(A,B,Xs,Ys2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys2)) ) ) ) ).

% hd_zip
tff(fact_5403_zip__same,axiom,
    ! [A: $tType,A3: A,B2: A,Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs))))
    <=> ( pp(aa(set(A),bool,member(A,A3),aa(list(A),set(A),set2(A),Xs)))
        & ( A3 = B2 ) ) ) ).

% zip_same
tff(fact_5404_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
     => ~ ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),Ys2))) ) ) ).

% in_set_zipE
tff(fact_5405_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_zip_leftD
tff(fact_5406_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
     => pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),Ys2))) ) ).

% set_zip_rightD
tff(fact_5407_rel__restrict__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : rel_restrict(A,R2,A5) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_abm(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),R2),A5))) ).

% rel_restrict_def
tff(fact_5408_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( zip(A,B,Xs,Ys2) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
     => ~ ! [X3: A,Xs5: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs5) )
           => ! [Y3: B,Ys5: list(B)] :
                ( ( Ys2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
                 => ( Xys != zip(A,B,Xs5,Ys5) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5409_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F3: fun(D,B),Xs: list(D),G3: fun(D,C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H)),zip(B,C,aa(list(D),list(B),map(D,B,F3),Xs),aa(list(D),list(C),map(D,C,G3),Xs))) = aa(list(D),list(A),map(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_abn(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F3),G3)),Xs) ).

% map2_map_map
tff(fact_5410_set__zip__cart,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),L: list(A),L3: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,L,L3))))
     => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_xy(list(B),fun(A,set(B)),L3)))) ) ).

% set_zip_cart
tff(fact_5411_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : zip(A,B,Xs,Ys2) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_wk(B,fun(A,product_prod(A,B))))),zip(B,A,Ys2,Xs)) ).

% zip_commute
tff(fact_5412_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys2)) ) ) ) ) ).

% last_zip
tff(fact_5413_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),Ys2)))
       => ~ ! [X3: A] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_5414_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ~ ! [Y3: B] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_5415_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
      <=> ? [Y5: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),X) = aa(B,option(B),some(B),Y5) ) ) ).

% map_of_zip_is_Some
tff(fact_5416_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M,Ks,Vs) = aa(list(product_prod(A,B)),fun(A,option(B)),aa(fun(A,option(B)),fun(list(product_prod(A,B)),fun(A,option(B))),foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_abp(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))))),M),zip(A,B,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_5417_rel__restrict__compl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: 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,R2,A5)),rel_restrict(A,R2,aa(set(A),set(A),uminus_uminus(set(A)),A5))) = bot_bot(set(product_prod(A,A))) ).

% rel_restrict_compl
tff(fact_5418_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(product_prod(B,C),A),G3: fun(D,B),Xs: list(D),Ys2: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,aa(list(D),list(B),map(D,B,G3),Xs),Ys2)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_abq(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F3),G3))),zip(D,C,Xs,Ys2)) ).

% map_zip_map
tff(fact_5419_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(product_prod(B,C),A),Xs: list(B),G3: fun(D,C),Ys2: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G3),Ys2))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_abr(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F3),G3))),zip(B,D,Xs,Ys2)) ).

% map_zip_map2
tff(fact_5420_foldl__snd__zip,axiom,
    ! [B: $tType,C: $tType,A: $tType,Ys2: list(A),Xs: list(B),F3: fun(C,fun(A,C)),B2: C] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(B),nat,size_size(list(B)),Xs)))
     => ( aa(list(product_prod(B,A)),C,aa(C,fun(list(product_prod(B,A)),C),foldl(C,product_prod(B,A),aTP_Lamp_abt(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F3)),B2),zip(B,A,Xs,Ys2)) = aa(list(A),C,aa(C,fun(list(A),C),foldl(C,A,F3),B2),Ys2) ) ) ).

% foldl_snd_zip
tff(fact_5421_homo__rel__restrict__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B4: set(A),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R2,A5)),product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5),aa(set(A),fun(A,set(A)),aTP_Lamp_abu(set(A),fun(set(A),fun(A,set(A))),B4),A5)))) ) ).

% homo_rel_restrict_mono
tff(fact_5422_rel__restrict__alt__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : rel_restrict(A,R2,A5) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(A),set(A),uminus_uminus(set(A)),A5),aTP_Lamp_abv(set(A),fun(A,set(A)),A5))) ).

% rel_restrict_alt_def
tff(fact_5423_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xs: list(C),Ys2: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),Ys2) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_abw(fun(C,A),fun(C,fun(B,product_prod(A,B))),F3))),zip(C,B,Xs,Ys2)) ).

% zip_map1
tff(fact_5424_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F3: fun(C,B),Ys2: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F3),Ys2)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_wi(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),zip(A,C,Xs,Ys2)) ).

% zip_map2
tff(fact_5425_map__prod__fun__zip,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),Xs: list(C),Ys2: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_yk(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),zip(C,D,Xs,Ys2)) = zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),aa(list(D),list(B),map(D,B,G3),Ys2)) ).

% map_prod_fun_zip
tff(fact_5426_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys2: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_abx(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys2)),Xs) ).

% zip_Cons
tff(fact_5427_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys2: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2) = aa(list(B),list(product_prod(A,B)),case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_aby(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys2) ).

% zip_Cons1
tff(fact_5428_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,B),X5: A] :
      ( ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X5) = aa(B,option(B),some(B),aa(A,B,F3,X5)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X5) = none(B) ) ) ) ).

% map_of_zip_map
tff(fact_5429_map__of__zip__upd,axiom,
    ! [A: $tType,B: $tType,Ys2: list(B),Xs: list(A),Zs: list(B),X: A,Y: B,Z2: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys2) = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
       => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),X,aa(B,option(B),some(B),Y)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X,aa(B,option(B),some(B),Z2)) )
           => ( map_of(A,B,zip(A,B,Xs,Ys2)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_5430_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P: fun(nat,bool),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_abz(fun(nat,bool),fun(product_prod(A,nat),bool),P)),zip(A,nat,Xs,Is))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_aca(fun(nat,bool),fun(product_prod(A,nat),bool),P)),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_5431_rel__restrict__Sigma__sub,axiom,
    ! [A: $tType,A5: set(A),R2: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,transitive_trancl(A,product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))),R2)),transitive_trancl(A,product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),R2),aa(set(A),fun(A,set(A)),aTP_Lamp_abu(set(A),fun(set(A),fun(A,set(A))),A5),R2))))) ).

% rel_restrict_Sigma_sub
tff(fact_5432_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),I2: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( distinct(A,Xs)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys2)))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),aa(nat,A,nth(A,Xs),I2)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),I2)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_5433_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),P3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
    <=> ? [N: nat] :
          ( ( aa(nat,A,nth(A,Xs),N) = aa(product_prod(A,B),A,product_fst(A,B),P3) )
          & ( aa(nat,B,nth(B,Ys2),N) = aa(product_prod(A,B),B,product_snd(A,B),P3) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Ys2))) ) ) ).

% in_set_zip
tff(fact_5434_nths__shift__lemma,axiom,
    ! [A: $tType,A5: set(nat),Xs: list(A),I2: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_acb(set(nat),fun(product_prod(A,nat),bool),A5)),zip(A,nat,Xs,upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_acc(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A5),I2)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_5435_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat)] : nths(A,Xs,A5) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_acb(set(nat),fun(product_prod(A,nat),bool),A5)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_5436_mset__zip__take__Cons__drop__twice,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),J: nat,X: A,Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( mset(product_prod(A,B),zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,J,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),drop(A,J,Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,J,Ys2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),drop(B,J,Ys2))))) = aa(multiset(product_prod(A,B)),multiset(product_prod(A,B)),aa(product_prod(A,B),fun(multiset(product_prod(A,B)),multiset(product_prod(A,B))),add_mset(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),mset(product_prod(A,B),zip(A,B,Xs,Ys2))) ) ) ) ).

% mset_zip_take_Cons_drop_twice
tff(fact_5437_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_abh(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) ) )
           => ~ ( ( X = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_5438_mergesort__by__rel__permutes,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] : mset(A,aa(list(A),list(A),mergesort_by_rel(A,R2),Xs)) = mset(A,Xs) ).

% mergesort_by_rel_permutes
tff(fact_5439_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_5440_quicksort__by__rel__permutes,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : mset(A,aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs)) = mset(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).

% quicksort_by_rel_permutes
tff(fact_5441_mset__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] : mset(A,merges9089515139780605204_merge(A,R2,Xs,Ys2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys2)) ).

% mset_mergesort_by_rel_merge
tff(fact_5442_Multiset_Omset__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : mset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),Xs)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),mset(A,Xs)) ) ).

% Multiset.mset_insort
tff(fact_5443_sorted__list__of__multiset__mset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : linord6283353356039996273ltiset(A,mset(A,Xs)) = aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),Xs) ) ).

% sorted_list_of_multiset_mset
tff(fact_5444_mset__mergesort__by__rel__split,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xs: list(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(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),Xs)))),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),Xs)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Xs12))),mset(A,Xs23)) ).

% mset_mergesort_by_rel_split
tff(fact_5445_mset__eq__finite,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_acd(list(A),fun(list(A),bool),Xs)))) ).

% mset_eq_finite
tff(fact_5446_mset__eq__length__filter,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Z2: A] :
      ( ( mset(A,Xs) = mset(A,Ys2) )
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),Z2)),Xs)) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),Z2)),Ys2)) ) ) ).

% mset_eq_length_filter
tff(fact_5447_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I5))),aa(list(A),set(A),set2(A),Xs))) ).

% set_nths_subset
tff(fact_5448_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(set(nat),bool,member(nat,I3),I5)) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_5449_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_5450_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_cq(A,A),Option) ).

% option.the_def
tff(fact_5451_drop__eq__nths,axiom,
    ! [A: $tType,N2: nat,Xs: list(A)] : drop(A,N2,Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less_eq(nat),N2))) ).

% drop_eq_nths
tff(fact_5452_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,case_list(A,A,undefined(A),aTP_Lamp_ace(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_5453_properties__for__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ys2: list(A),Xs: list(A)] :
          ( ( mset(A,Ys2) = mset(A,Xs) )
         => ( sorted_wrt(A,ord_less_eq(A),Ys2)
           => ( aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),Xs) = Ys2 ) ) ) ) ).

% properties_for_sort
tff(fact_5454_mset__swap,axiom,
    ! [A: $tType,I2: nat,Ls2: list(A),J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ls2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Ls2)))
       => ( mset(A,list_update(A,list_update(A,Ls2,J,aa(nat,A,nth(A,Ls2),I2)),I2,aa(nat,A,nth(A,Ls2),J))) = mset(A,Ls2) ) ) ) ).

% mset_swap
tff(fact_5455_nths__append,axiom,
    ! [A: $tType,L: list(A),L3: list(A),A5: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L3),A5) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A5)),nths(A,L3,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_acf(list(A),fun(set(nat),fun(nat,bool)),L),A5)))) ).

% nths_append
tff(fact_5456_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S2: set(nat)] :
      ( distinct(A,Xs)
     => ( aa(list(A),list(A),filter2(A,aa(set(nat),fun(A,bool),aTP_Lamp_acg(list(A),fun(set(nat),fun(A,bool)),Xs),S2)),Xs) = nths(A,Xs,S2) ) ) ).

% filter_in_nths
tff(fact_5457_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ach(list(A),fun(set(nat),fun(nat,bool)),Xs),I5))) ).

% length_nths
tff(fact_5458_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_ss(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).

% filter_eq_nths
tff(fact_5459_properties__for__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Ys2: list(B),Xs: list(B),F3: fun(B,A)] :
          ( ( mset(B,Ys2) = mset(B,Xs) )
         => ( ! [K2: B] :
                ( pp(aa(set(B),bool,member(B,K2),aa(list(B),set(B),set2(B),Ys2)))
               => ( aa(list(B),list(B),filter2(B,aa(B,fun(B,bool),aTP_Lamp_aci(fun(B,A),fun(B,fun(B,bool)),F3),K2)),Ys2) = aa(list(B),list(B),filter2(B,aa(B,fun(B,bool),aTP_Lamp_aci(fun(B,A),fun(B,fun(B,bool)),F3),K2)),Xs) ) )
           => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Ys2))
             => ( aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = Ys2 ) ) ) ) ) ).

% properties_for_sort_key
tff(fact_5460_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A5: set(nat)] : nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),A5) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),if(list(A),aa(set(nat),bool,member(nat,zero_zero(nat)),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),nil(A))),nths(A,L,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_acj(set(nat),fun(nat,bool),A5)))) ).

% nths_Cons
tff(fact_5461_mset__update,axiom,
    ! [A: $tType,I2: nat,Ls2: list(A),V2: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ls2)))
     => ( mset(A,list_update(A,Ls2,I2,V2)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),V2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),mset(A,Ls2)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),aa(nat,A,nth(A,Ls2),I2)),zero_zero(multiset(A))))) ) ) ).

% mset_update
tff(fact_5462_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = aa(fun(A,bool),set(A),collect(A),aa(set(nat),fun(A,bool),aTP_Lamp_ack(list(A),fun(set(nat),fun(A,bool)),Xs),I5)) ).

% set_nths
tff(fact_5463_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),X))
           => ( ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_abh(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) )
                   => ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) )
             => ~ ( ( X = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),nil(A))) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_5464_mod__h__bot__normalize,axiom,
    ! [A: $tType,H: heap_ext(product_unit),P: assn] :
      ( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
      <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_normalize
tff(fact_5465_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa2: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa2) = Y )
         => ( ! [X3: A] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( Y != X3 ) )
           => ( ! [X3: A,Y3: A,Zs2: list(A)] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                 => ( Y != if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))) ) )
             => ~ ( ( Xa2 = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_5466_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))),X,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) ) ).

% arg_min_list.simps(2)
tff(fact_5467_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa2: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa2) = Y )
         => ( pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),Xa2)))
           => ( ! [X3: A] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                 => ( ( Y = X3 )
                   => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) )
             => ( ! [X3: A,Y3: A,Zs2: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                   => ( ( Y = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))) )
                     => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))))) ) )
               => ~ ( ( Xa2 = nil(A) )
                   => ( ( Y = undefined(A) )
                     => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),nil(A)))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_5468_mult_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [X: A,Y: A,A3: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ) ) ).

% mult.safe_commute
tff(fact_5469_image__mset__map__of,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(multiset(A),multiset(B),image_mset(A,B,aTP_Lamp_acl(list(product_prod(A,B)),fun(A,B),Xs)),mset(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))) = mset(B,aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs)) ) ) ).

% image_mset_map_of
tff(fact_5470_multiset_Omap__ident,axiom,
    ! [A: $tType,T6: multiset(A)] : aa(multiset(A),multiset(A),image_mset(A,A,aTP_Lamp_cq(A,A)),T6) = T6 ).

% multiset.map_ident
tff(fact_5471_mset__map__split__orig,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),P: multiset(B),M12: multiset(A),M23: multiset(A)] :
      ( ( aa(multiset(B),multiset(A),image_mset(B,A,F3),P) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M12),M23) )
     => ~ ! [P12: multiset(B),P22: multiset(B)] :
            ( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P22) )
           => ( ( aa(multiset(B),multiset(A),image_mset(B,A,F3),P12) = M12 )
             => ( aa(multiset(B),multiset(A),image_mset(B,A,F3),P22) != M23 ) ) ) ) ).

% mset_map_split_orig
tff(fact_5472_mset__map__id,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),G3: fun(A,B),X6: multiset(A)] :
      ( ! [X3: A] : aa(B,A,F3,aa(A,B,G3,X3)) = X3
     => ( aa(multiset(B),multiset(A),image_mset(B,A,F3),aa(multiset(A),multiset(B),image_mset(A,B,G3),X6)) = X6 ) ) ).

% mset_map_id
tff(fact_5473_mset__map__split__orig__le,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),P: multiset(B),M12: multiset(A),M23: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),P)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M12),M23)))
     => ~ ! [P12: multiset(B),P22: multiset(B)] :
            ( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P22) )
           => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),P12)),M12))
             => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),P22)),M23)) ) ) ) ).

% mset_map_split_orig_le
tff(fact_5474_mult_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.right_assoc
tff(fact_5475_mult_Oright__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ).

% mult.right_commute
tff(fact_5476_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),A5: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_acm(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3)),A5) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F3),A5),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_acn(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F3),G3),A5)) ).

% image_split_eq_Sigma
tff(fact_5477_enumerate__replicate__eq,axiom,
    ! [A: $tType,N2: nat,M: nat,A3: A] : enumerate(A,N2,replicate(A,M,A3)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_aco(A,fun(nat,product_prod(nat,A)),A3)),upt(N2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M))) ).

% enumerate_replicate_eq
tff(fact_5478_subset__eq__mset__impl,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( ( subset_eq_mset_impl(A,Xs,Ys2) = none(bool) )
      <=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),mset(A,Xs)),mset(A,Ys2))) )
      & ( ( subset_eq_mset_impl(A,Xs,Ys2) = aa(bool,option(bool),some(bool),fTrue) )
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),mset(A,Xs)),mset(A,Ys2))) )
      & ( ( subset_eq_mset_impl(A,Xs,Ys2) = aa(bool,option(bool),some(bool),fFalse) )
       => ( mset(A,Xs) = mset(A,Ys2) ) ) ) ).

% subset_eq_mset_impl
tff(fact_5479_vimage__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),B4: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)))
    <=> pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),B4)) ) ).

% vimage_eq
tff(fact_5480_vimageI,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: B,B2: A,B4: set(A)] :
      ( ( aa(B,A,F3,A3) = B2 )
     => ( pp(aa(set(A),bool,member(A,B2),B4))
       => pp(aa(set(B),bool,member(B,A3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B4))) ) ) ).

% vimageI
tff(fact_5481_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),P: fun(B,bool)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(fun(B,bool),set(B),collect(B),P)) = aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_acp(fun(A,B),fun(fun(B,bool),fun(A,bool)),F3),P)) ).

% vimage_Collect_eq
tff(fact_5482_vimage__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_cq(A,A)),Y6) = Y6 ).

% vimage_ident
tff(fact_5483_vimage__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),bot_bot(set(B))) = bot_bot(set(A)) ).

% vimage_empty
tff(fact_5484_vimage__UNIV,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),top_top(set(B))) = top_top(set(A)) ).

% vimage_UNIV
tff(fact_5485_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B),B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)) ).

% vimage_Int
tff(fact_5486_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B),B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)) ).

% vimage_Un
tff(fact_5487_nth__replicate,axiom,
    ! [A: $tType,I2: nat,N2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N2))
     => ( aa(nat,A,nth(A,replicate(A,N2,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_5488_vimage__const,axiom,
    ! [B: $tType,A: $tType,C2: B,A5: set(B)] :
      ( ( pp(aa(set(B),bool,member(B,C2),A5))
       => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aTP_Lamp_qf(B,fun(A,B),C2)),A5) = top_top(set(A)) ) )
      & ( ~ pp(aa(set(B),bool,member(B,C2),A5))
       => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aTP_Lamp_qf(B,fun(A,B),C2)),A5) = bot_bot(set(A)) ) ) ) ).

% vimage_const
tff(fact_5489_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(A)] : aa(set(B),set(A),image2(B,A,F3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))) ).

% image_vimage_eq
tff(fact_5490_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I2: nat,X: A,J: nat,Y: B] : zip(A,B,replicate(A,I2,X),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_replicate
tff(fact_5491_vimage__if,axiom,
    ! [B: $tType,A: $tType,C2: B,A5: set(B),D3: B,B4: set(A)] :
      ( ( pp(aa(set(B),bool,member(B,C2),A5))
       => ( ( pp(aa(set(B),bool,member(B,D3),A5))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_acq(B,fun(B,fun(set(A),fun(A,B))),C2),D3),B4)),A5) = top_top(set(A)) ) )
          & ( ~ pp(aa(set(B),bool,member(B,D3),A5))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_acq(B,fun(B,fun(set(A),fun(A,B))),C2),D3),B4)),A5) = B4 ) ) ) )
      & ( ~ pp(aa(set(B),bool,member(B,C2),A5))
       => ( ( pp(aa(set(B),bool,member(B,D3),A5))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_acq(B,fun(B,fun(set(A),fun(A,B))),C2),D3),B4)),A5) = aa(set(A),set(A),uminus_uminus(set(A)),B4) ) )
          & ( ~ pp(aa(set(B),bool,member(B,D3),A5))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_acq(B,fun(B,fun(set(A),fun(A,B))),C2),D3),B4)),A5) = bot_bot(set(A)) ) ) ) ) ) ).

% vimage_if
tff(fact_5492_set__replicate,axiom,
    ! [A: $tType,N2: nat,X: A] :
      ( ( N2 != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,N2,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_5493_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_fst_mk_fst
tff(fact_5494_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(B),list(product_prod(B,A)),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_wx(A,fun(B,product_prod(B,A))),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_snd_mk_snd
tff(fact_5495_subset__mset_Olexordp_Omono,axiom,
    ! [A: $tType] : pp(aa(fun(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))),bool,order_mono(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))),aTP_Lamp_acr(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))))) ).

% subset_mset.lexordp.mono
tff(fact_5496_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),B2: B] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B2),bot_bot(set(B))))))
    <=> ( aa(A,B,F3,A3) = B2 ) ) ).

% vimage_singleton_eq
tff(fact_5497_image__vimage__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A5))),A5)) ).

% image_vimage_subset
tff(fact_5498_image__subset__iff__subset__vimage,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A5)),B4))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B4))) ) ).

% image_subset_iff_subset_vimage
tff(fact_5499_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),uminus_uminus(set(B)),A5)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)) ).

% vimage_Compl
tff(fact_5500_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B),B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)) ).

% vimage_Diff
tff(fact_5501_vimage__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),F3: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A5)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B4))) ) ).

% vimage_mono
tff(fact_5502_subset__vimage__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,B),B4: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(set(B),bool,member(B,aa(A,B,F3,X4)),B4)) ) ) ).

% subset_vimage_iff
tff(fact_5503_vimage__Collect,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F3: fun(A,B),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(B,bool,P,aa(A,B,F3,X3)))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(fun(B,bool),set(B),collect(B),P)) = aa(fun(A,bool),set(A),collect(A),Q) ) ) ).

% vimage_Collect
tff(fact_5504_vimageI2,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: B,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,aa(B,A,F3,A3)),A5))
     => pp(aa(set(B),bool,member(B,A3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A5))) ) ).

% vimageI2
tff(fact_5505_vimageE,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),B4: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)))
     => pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),B4)) ) ).

% vimageE
tff(fact_5506_vimageD,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)))
     => pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),A5)) ) ).

% vimageD
tff(fact_5507_vimage__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_acs(fun(A,B),fun(set(B),fun(A,bool)),F3),B4)) ).

% vimage_def
tff(fact_5508_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,B),G3: fun(A,B),Y: set(B)] :
      ( ! [W: A] :
          ( pp(aa(set(A),bool,member(A,W),S))
         => ( aa(A,B,F3,W) = aa(A,B,G3,W) ) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),Y)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),G3),Y)),S) ) ) ).

% vimage_inter_cong
tff(fact_5509_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N2: nat,X: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,N2,X)) ) ).

% sorted_replicate
tff(fact_5510_subset__mset_Olift__Suc__mono__less,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N2: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N5)),aa(nat,multiset(A),F3,aa(nat,nat,suc,N5))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N6))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N2)),aa(nat,multiset(A),F3,N6))) ) ) ).

% subset_mset.lift_Suc_mono_less
tff(fact_5511_subset__mset_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N2: nat,M: nat] :
      ( ! [N5: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N5)),aa(nat,multiset(A),F3,aa(nat,nat,suc,N5))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N2)),aa(nat,multiset(A),F3,M)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M)) ) ) ).

% subset_mset.lift_Suc_mono_less_iff
tff(fact_5512_mset__subset__size,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A5),B4))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),A5)),aa(multiset(A),nat,size_size(multiset(A)),B4))) ) ).

% mset_subset_size
tff(fact_5513_vimage__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),B4: fun(C,set(B)),A5: set(C)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_act(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F3),B4)),A5)) ).

% vimage_UN
tff(fact_5514_vimage__INT,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),B4: fun(C,set(B)),A5: set(C)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_act(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F3),B4)),A5)) ).

% vimage_INT
tff(fact_5515_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_px(A,fun(B,A)),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_5516_vimage__Times,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,product_prod(B,C)),A5: set(B),B4: set(C)] : aa(set(product_prod(B,C)),set(A),aa(fun(A,product_prod(B,C)),fun(set(product_prod(B,C)),set(A)),vimage(A,product_prod(B,C)),F3),product_Sigma(B,C,A5,aTP_Lamp_yg(set(C),fun(B,set(C)),B4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F3)),A5)),aa(set(C),set(A),aa(fun(A,C),fun(set(C),set(A)),vimage(A,C),aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F3)),B4)) ).

% vimage_Times
tff(fact_5517_vimage__Union,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(set(B))] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3)),A5)) ).

% vimage_Union
tff(fact_5518_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),X)),Xs)),X) = aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),X)),Xs) ).

% replicate_length_filter
tff(fact_5519_subset__mset_Oordering__top__axioms,axiom,
    ! [A: $tType] : ordering_top(multiset(A),aTP_Lamp_acu(multiset(A),fun(multiset(A),bool)),aTP_Lamp_acv(multiset(A),fun(multiset(A),bool)),zero_zero(multiset(A))) ).

% subset_mset.ordering_top_axioms
tff(fact_5520_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,X: B,A5: set(B),F3: fun(B,set(A))] :
      ( ( pp(aa(set(B),bool,member(B,X),A5))
       => ( aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X)),product_Sigma(B,A,A5,F3)) = aa(B,set(A),F3,X) ) )
      & ( ~ pp(aa(set(B),bool,member(B,X),A5))
       => ( aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X)),product_Sigma(B,A,A5,F3)) = bot_bot(set(A)) ) ) ) ).

% Pair_vimage_Sigma
tff(fact_5521_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => ( ( aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A5) = bot_bot(set(B)) )
      <=> ( A5 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_5522_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),B4: set(A),A5: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B4)),A5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image2(B,A,F3),A5))) ) ) ).

% vimage_subsetD
tff(fact_5523_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A3: B,B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)) ).

% vimage_insert
tff(fact_5524_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N2: nat,C2: A] : groups8242544230860333062m_list(A,replicate(A,N2,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),C2) ) ).

% sum_list_replicate
tff(fact_5525_vimage__fst,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : aa(set(A),set(product_prod(A,B)),aa(fun(product_prod(A,B),A),fun(set(A),set(product_prod(A,B))),vimage(product_prod(A,B),A),product_fst(A,B)),A5) = product_Sigma(A,B,A5,aTP_Lamp_xu(A,set(B))) ).

% vimage_fst
tff(fact_5526_vimage__snd,axiom,
    ! [A: $tType,B: $tType,A5: set(B)] : aa(set(B),set(product_prod(A,B)),aa(fun(product_prod(A,B),B),fun(set(B),set(product_prod(A,B))),vimage(product_prod(A,B),B),product_snd(A,B)),A5) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xt(set(B),fun(A,set(B)),A5)) ).

% vimage_snd
tff(fact_5527_size__psubset,axiom,
    ! [A: $tType,M6: multiset(A),M9: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M6),M9))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),M6)),aa(multiset(A),nat,size_size(multiset(A)),M9)))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),M6),M9)) ) ) ).

% size_psubset
tff(fact_5528_inf__img__fin__domE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A5))
       => ~ ! [Y3: A] :
              ( pp(aa(set(A),bool,member(A,Y3),aa(set(B),set(A),image2(B,A,F3),A5)))
             => pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))) ) ) ) ).

% inf_img_fin_domE
tff(fact_5529_inf__img__fin__dom,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A5))
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(set(B),set(A),image2(B,A,F3),A5)))
            & ~ pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A)))))) ) ) ) ).

% inf_img_fin_dom
tff(fact_5530_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),aa(set(A),set(B),image2(A,B,F3),top_top(set(A)))))
       => pp(aa(set(B),bool,finite_finite2(B),A5)) ) ) ).

% finite_vimageD'
tff(fact_5531_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I2: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_acw(A,fun(nat,A),X)),upt(zero_zero(nat),I2)) = replicate(A,I2,X) ).

% map_replicate_trivial
tff(fact_5532_finite__finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( ! [Y3: A] :
            ( pp(aa(set(A),bool,member(A,Y3),F4))
           => pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))),A5))) )
       => pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H),F4)),A5))) ) ) ).

% finite_finite_vimage_IntI
tff(fact_5533_set__replicate__conv__if,axiom,
    ! [A: $tType,N2: nat,X: A] :
      ( ( ( N2 = zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N2,X)) = bot_bot(set(A)) ) )
      & ( ( N2 != zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N2,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ).

% set_replicate_conv_if
tff(fact_5534_set__replicate__Suc,axiom,
    ! [A: $tType,N2: nat,X: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,N2),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_5535_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,N2: nat,X: A] : replicate(A,aa(nat,nat,suc,N2),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N2,X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% replicate_Suc_conv_snoc
tff(fact_5536_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_acx(fun(A,B),fun(B,set(A)),F3)),B4)) ).

% vimage_eq_UN
tff(fact_5537_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N2: nat,X: A,Ys2: list(B)] : zip(A,B,replicate(A,N2,X),Ys2) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),take(B,N2,Ys2)) ).

% zip_replicate1
tff(fact_5538_inf__img__fin__dom_H,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A5))
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(set(B),set(A),image2(B,A,F3),A5)))
            & ~ pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))),A5))) ) ) ) ).

% inf_img_fin_dom'
tff(fact_5539_inf__img__fin__domE_H,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A5))
       => ~ ! [Y3: A] :
              ( pp(aa(set(A),bool,member(A,Y3),aa(set(B),set(A),image2(B,A,F3),A5)))
             => pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))),A5))) ) ) ) ).

% inf_img_fin_domE'
tff(fact_5540_map__zip2,axiom,
    ! [A: $tType,B: $tType,K: A,L: list(B)] : aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K),L) ).

% map_zip2
tff(fact_5541_map__zip1,axiom,
    ! [A: $tType,B: $tType,K: B,L: list(A)] : aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_wk(B,fun(A,product_prod(A,B))),K)),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K)) ).

% map_zip1
tff(fact_5542_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),N2: nat,Y: B] : zip(A,B,Xs,replicate(B,N2,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_wk(B,fun(A,product_prod(A,B))),Y)),take(A,N2,Xs)) ).

% zip_replicate2
tff(fact_5543_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N2: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = replicate(A,N2,Y) )
    <=> ( ( X = Y )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_5544_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Xs) )
         => ? [N5: nat,Zs2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N5))
              & ( concat(A,replicate(list(A),N5,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_5545_subset__mset_Osum__pos,axiom,
    ! [A: $tType,B: $tType,I5: set(B),F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),I5))
     => ( ( I5 != bot_bot(set(B)) )
       => ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),I5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I3))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,I5))) ) ) ) ).

% subset_mset.sum_pos
tff(fact_5546_subset__mset_Osum__strict__mono,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),A5))
     => ( ( A5 != bot_bot(set(B)) )
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(B,multiset(A),F3,X3)),aa(B,multiset(A),G3,X3))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,A5)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),G3,A5))) ) ) ) ).

% subset_mset.sum_strict_mono
tff(fact_5547_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys2: list(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
     => ( ! [Y3: A] :
            ( pp(aa(set(A),bool,member(A,Y3),aa(list(A),set(A),set2(A),Ys2)))
           => ~ pp(aa(A,bool,P,Y3)) )
       => ( aa(set(product_prod(list(A),list(A))),set(list(A)),aa(fun(list(A),product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(list(A))),vimage(list(A),product_prod(list(A),list(A))),partition(A,P)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),insert(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys2) ) ) ) ).

% inv_image_partition
tff(fact_5548_subset__mset_OgreaterThanLessThan__empty,axiom,
    ! [A: $tType,L: multiset(A),K: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),L),K))
     => ( set_gr287244882034783167ssThan(multiset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.greaterThanLessThan_empty
tff(fact_5549_subset__mset_OgreaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType,K: multiset(A),L: multiset(A)] :
      ( ( bot_bot(set(multiset(A))) = set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) )
    <=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),K),L)) ) ).

% subset_mset.greaterThanAtMost_empty_iff2
tff(fact_5550_subset__mset_OgreaterThanAtMost__empty__iff,axiom,
    ! [A: $tType,K: multiset(A),L: multiset(A)] :
      ( ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) )
    <=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),K),L)) ) ).

% subset_mset.greaterThanAtMost_empty_iff
tff(fact_5551_subset__mset_OgreaterThanAtMost__empty,axiom,
    ! [A: $tType,L: multiset(A),K: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),L),K))
     => ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.greaterThanAtMost_empty
tff(fact_5552_subset__mset_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),bot_bot(set(multiset(A))))) ).

% subset_mset.greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5553_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R2: set(C)] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G3),top_top(set(C)))),S))
       => finite673082921795544331dem_on(C,B,R2,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_5554_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_rc(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_5555_image__mset_Oidentity,axiom,
    ! [A: $tType] : image_mset(A,A,aTP_Lamp_cq(A,A)) = id(multiset(A)) ).

% image_mset.identity
tff(fact_5556_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_5557_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_5558_id__funpow,axiom,
    ! [A: $tType,N2: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),id(A)) = id(A) ).

% id_funpow
tff(fact_5559_subset__mset_OatLeastatMost__subset__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A),D3: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),set_atLeastAtMost(multiset(A),subseteq_mset(A),C2,D3)))
    <=> ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
        | ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),A3))
          & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),D3)) ) ) ) ).

% subset_mset.atLeastatMost_subset_iff
tff(fact_5560_apfst__id,axiom,
    ! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).

% apfst_id
tff(fact_5561_apsnd__id,axiom,
    ! [B: $tType,A: $tType] : aa(fun(B,B),fun(product_prod(A,B),product_prod(A,B)),product_apsnd(B,B,A),id(B)) = id(product_prod(A,B)) ).

% apsnd_id
tff(fact_5562_subset__mset_OatLeastatMost__empty__iff2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( bot_bot(set(multiset(A))) = set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) )
    <=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% subset_mset.atLeastatMost_empty_iff2
tff(fact_5563_subset__mset_OatLeastatMost__empty__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = bot_bot(set(multiset(A))) )
    <=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% subset_mset.atLeastatMost_empty_iff
tff(fact_5564_subset__mset_OatLeastatMost__empty,axiom,
    ! [A: $tType,B2: multiset(A),A3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),A3))
     => ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.atLeastatMost_empty
tff(fact_5565_comp__the__Some,axiom,
    ! [A: $tType] : aa(fun(A,option(A)),fun(A,A),comp(option(A),A,A,the2(A)),some(A)) = id(A) ).

% comp_the_Some
tff(fact_5566_subset__mset_OatLeastAtMost__singleton,axiom,
    ! [A: $tType,A3: multiset(A)] : set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,A3) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),bot_bot(set(multiset(A)))) ).

% subset_mset.atLeastAtMost_singleton
tff(fact_5567_subset__mset_OatLeastAtMost__singleton__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
      ( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),C2),bot_bot(set(multiset(A)))) )
    <=> ( ( A3 = B2 )
        & ( B2 = C2 ) ) ) ).

% subset_mset.atLeastAtMost_singleton_iff
tff(fact_5568_set_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_cq(A,A)) = id(set(A)) ).

% set.identity
tff(fact_5569_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_cq(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_5570_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))) ).

% nat_def
tff(fact_5571_map__option_Oidentity,axiom,
    ! [A: $tType] : map_option(A,A,aTP_Lamp_cq(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_5572_map__fun_Oidentity,axiom,
    ! [B: $tType,A: $tType] : map_fun(A,A,B,B,aTP_Lamp_cq(A,A),aTP_Lamp_aay(B,B)) = id(fun(A,B)) ).

% map_fun.identity
tff(fact_5573_option_Omap__id0,axiom,
    ! [A: $tType] : map_option(A,A,id(A)) = id(option(A)) ).

% option.map_id0
tff(fact_5574_option_Omap__id,axiom,
    ! [A: $tType,T6: option(A)] : aa(option(A),option(A),map_option(A,A,id(A)),T6) = T6 ).

% option.map_id
tff(fact_5575_map__prod_Oidentity,axiom,
    ! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_cq(A,A),aTP_Lamp_aay(B,B)) = id(product_prod(A,B)) ).

% map_prod.identity
tff(fact_5576_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3) = id(A) ).

% funpow_simps_right(1)
tff(fact_5577_apfst__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,C)] : product_apfst(A,C,B,F3) = product_map_prod(A,C,B,B,F3,id(B)) ).

% apfst_def
tff(fact_5578_apsnd__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] : aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3) = product_map_prod(A,A,B,C,id(A),F3) ).

% apsnd_def
tff(fact_5579_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_re(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_int_def
tff(fact_5580_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rg(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_eq_int_def
tff(fact_5581_subset__mset_OatLeastatMost__psubset__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A),D3: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),set_atLeastAtMost(multiset(A),subseteq_mset(A),C2,D3)))
    <=> ( ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
          | ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),A3))
            & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),D3))
            & ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),C2),A3))
              | pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),D3)) ) ) )
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),D3)) ) ) ).

% subset_mset.atLeastatMost_psubset_iff
tff(fact_5582_fst__diag__id,axiom,
    ! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_fst(A,A)),aTP_Lamp_uv(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).

% fst_diag_id
tff(fact_5583_snd__diag__id,axiom,
    ! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_snd(A,A)),aTP_Lamp_uv(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).

% snd_diag_id
tff(fact_5584_subset__mset_OatLeastAtMost__singleton_H,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( A3 = B2 )
     => ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),bot_bot(set(multiset(A)))) ) ) ).

% subset_mset.atLeastAtMost_singleton'
tff(fact_5585_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A5: set(A),Z2: B] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z2,A5)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_5586_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A5: set(A),Z2: B] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( finite_fold(A,B,F3,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z2),A5) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_5587_subset__mset_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),B2),bot_bot(set(multiset(A))))) ).

% subset_mset.atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5588_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),bool),aTP_Lamp_zu(fun(set(A),bool),fun(set(A),bool))) ).

% finite_def
tff(fact_5589_revg__fun,axiom,
    ! [A: $tType,A3: list(A),B2: list(A)] : revg(A,A3,B2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,A3)),B2) ).

% revg_fun
tff(fact_5590_subset__mset_OatLeastLessThan__empty,axiom,
    ! [A: $tType,B2: multiset(A),A3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),A3))
     => ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = bot_bot(set(multiset(A))) ) ) ).

% subset_mset.atLeastLessThan_empty
tff(fact_5591_subset__mset_OatLeastLessThan__empty__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = bot_bot(set(multiset(A))) )
    <=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2)) ) ).

% subset_mset.atLeastLessThan_empty_iff
tff(fact_5592_subset__mset_OatLeastLessThan__empty__iff2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( ( bot_bot(set(multiset(A))) = set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) )
    <=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2)) ) ).

% subset_mset.atLeastLessThan_empty_iff2
tff(fact_5593_less__eq__integer__def,axiom,
    ord_less_eq(code_integer) = aa(fun(int,fun(int,bool)),fun(code_integer,fun(code_integer,bool)),map_fun(code_integer,int,fun(int,bool),fun(code_integer,bool),code_int_of_integer,map_fun(code_integer,int,bool,bool,code_int_of_integer,id(bool))),ord_less_eq(int)) ).

% less_eq_integer_def
tff(fact_5594_less__integer__def,axiom,
    ord_less(code_integer) = aa(fun(int,fun(int,bool)),fun(code_integer,fun(code_integer,bool)),map_fun(code_integer,int,fun(int,bool),fun(code_integer,bool),code_int_of_integer,map_fun(code_integer,int,bool,bool,code_int_of_integer,id(bool))),ord_less(int)) ).

% less_integer_def
tff(fact_5595_lfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T6: A] : complete_lattice_lfp(A,aTP_Lamp_acy(A,fun(A,A),T6)) = T6 ) ).

% lfp_const
tff(fact_5596_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A5: A] :
          ( ! [U5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,U5)),U5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),U5)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),complete_lattice_lfp(A,F3))) ) ) ).

% lfp_greatest
tff(fact_5597_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,A5)),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),A5)) ) ) ).

% lfp_lowerbound
tff(fact_5598_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),G3: fun(A,A)] :
          ( ! [Z8: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,Z8)),aa(A,A,G3,Z8)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),complete_lattice_lfp(A,G3))) ) ) ).

% lfp_mono
tff(fact_5599_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,fun(A,A))] :
          ( ! [X3: A,Y3: A,W: A,Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),W)),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_acz(fun(A,fun(A,A)),fun(A,A),F3)) = complete_lattice_lfp(A,aTP_Lamp_ada(fun(A,fun(A,A)),fun(A,A),F3)) ) ) ) ).

% lfp_lfp
tff(fact_5600_lfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G3: fun(A,B),F3: fun(B,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),G3))
         => ( pp(aa(fun(B,A),bool,order_mono(B,A),F3))
           => ( aa(A,B,G3,complete_lattice_lfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_adb(fun(A,B),fun(fun(B,A),fun(A,A)),G3),F3))) = complete_lattice_lfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_adc(fun(A,B),fun(fun(B,A),fun(B,B)),G3),F3)) ) ) ) ) ).

% lfp_rolling
tff(fact_5601_revg_Osimps_I2_J,axiom,
    ! [A: $tType,A3: A,As: list(A),B2: list(A)] : revg(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As),B2) = revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),B2)) ).

% revg.simps(2)
tff(fact_5602_revg_Osimps_I1_J,axiom,
    ! [A: $tType,B2: list(A)] : revg(A,nil(A),B2) = B2 ).

% revg.simps(1)
tff(fact_5603_lfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),X: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F4))
         => ( ( aa(A,A,F4,X) = X )
           => ( ! [Z3: A] :
                  ( ( aa(A,A,F4,Z3) = Z3 )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z3)) )
             => ( complete_lattice_lfp(A,F4) = X ) ) ) ) ) ).

% lfp_eqI
tff(fact_5604_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] : complete_lattice_lfp(A,F3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_add(fun(A,A),fun(A,bool),F3))) ) ).

% lfp_def
tff(fact_5605_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: A,F3: fun(A,A),P: A] :
          ( ( A5 = complete_lattice_lfp(A,F3) )
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),P))),P))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),P)) ) ) ) ) ).

% def_lfp_induct
tff(fact_5606_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F3)),P))),P))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),P)) ) ) ) ).

% lfp_induct
tff(fact_5607_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P: fun(A,bool)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ! [S8: A] :
                ( pp(aa(A,bool,P,S8))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S8),complete_lattice_lfp(A,F3)))
                 => pp(aa(A,bool,P,aa(A,A,F3,S8))) ) )
           => ( ! [M11: set(A)] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),M11))
                     => pp(aa(A,bool,P,X5)) )
                 => pp(aa(A,bool,P,aa(set(A),A,complete_Sup_Sup(A),M11))) )
             => pp(aa(A,bool,P,complete_lattice_lfp(A,F3))) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_5608_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N2)),F3)) = complete_lattice_lfp(A,F3) ) ) ) ).

% lfp_funpow
tff(fact_5609_revg_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: list(A)] :
      ( ( revg(A,X,Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != Xa2 ) )
       => ~ ! [A6: A,As4: list(A)] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
             => ( Y != revg(A,As4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),Xa2)) ) ) ) ) ).

% revg.elims
tff(fact_5610_subset__mset_OatLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A),D3: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),C2,D3)))
    <=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),A3))
          & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),D3)) ) ) ) ).

% subset_mset.atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5611_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F3),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F3),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F3),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_5612_revg_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: list(A)] :
      ( ( revg(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa2)))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa2 )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,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)),Xa2))) ) )
         => ~ ! [A6: A,As4: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
               => ( ( Y = revg(A,As4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),Xa2)) )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),Xa2))) ) ) ) ) ) ).

% revg.pelims
tff(fact_5613_subset__mset_OIcc__subset__Iic__iff,axiom,
    ! [A: $tType,L: multiset(A),H: multiset(A),H3: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),L,H)),set_atMost(multiset(A),subseteq_mset(A),H3)))
    <=> ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),L),H))
        | pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),H),H3)) ) ) ).

% subset_mset.Icc_subset_Iic_iff
tff(fact_5614_swap__comp__swap,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B)),comp(product_prod(B,A),product_prod(A,B),product_prod(A,B),product_swap(B,A)),product_swap(A,B)) = id(product_prod(A,B)) ).

% swap_comp_swap
tff(fact_5615_swap__swap,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P3)) = P3 ).

% swap_swap
tff(fact_5616_swap__simp,axiom,
    ! [A: $tType,B: $tType,X: B,Y: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X) ).

% swap_simp
tff(fact_5617_case__swap,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,fun(B,A)),P3: product_prod(C,B)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aTP_Lamp_ade(fun(C,fun(B,A)),fun(B,fun(C,A)),F3)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P3)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F3),P3) ).

% case_swap
tff(fact_5618_fst__swap,axiom,
    ! [A: $tType,B: $tType,X: product_prod(B,A)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),X)) = aa(product_prod(B,A),A,product_snd(B,A),X) ).

% fst_swap
tff(fact_5619_snd__swap,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),X)) = aa(product_prod(A,B),A,product_fst(A,B),X) ).

% snd_swap
tff(fact_5620_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Y: A,X: B,A5: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X)),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),A5)))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)),A5)) ) ).

% pair_in_swap_image
tff(fact_5621_surj__swap,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).

% surj_swap
tff(fact_5622_ord_OatMost__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),U: A] : set_atMost(A,Less_eq,U) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),U)) ).

% ord.atMost_def
tff(fact_5623_subset__mset_OatMost__def,axiom,
    ! [A: $tType,U: multiset(A)] : set_atMost(multiset(A),subseteq_mset(A),U) = aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_acu(multiset(A),fun(multiset(A),bool)),U)) ).

% subset_mset.atMost_def
tff(fact_5624_def__lfp__induct__set,axiom,
    ! [A: $tType,A5: set(A),F3: fun(set(A),set(A)),A3: A,P: fun(A,bool)] :
      ( ( A5 = complete_lattice_lfp(set(A),F3) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,bool),set(A),collect(A),P)))))
               => pp(aa(A,bool,P,X3)) )
           => pp(aa(A,bool,P,A3)) ) ) ) ) ).

% def_lfp_induct_set
tff(fact_5625_lfp__induct__set,axiom,
    ! [A: $tType,A3: A,F3: fun(set(A),set(A)),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,A3),complete_lattice_lfp(set(A),F3)))
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_lattice_lfp(set(A),F3)),aa(fun(A,bool),set(A),collect(A),P)))))
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% lfp_induct_set
tff(fact_5626_subset__mset_Onot__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType,H: multiset(A)] : bot_bot(set(multiset(A))) != set_atMost(multiset(A),subseteq_mset(A),H) ).

% subset_mset.not_empty_eq_Iic_eq_empty
tff(fact_5627_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,F3: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),complete_lattice_lfp(set(product_prod(A,B)),F3)))
     => ( pp(aa(fun(set(product_prod(A,B)),set(product_prod(A,B))),bool,order_mono(set(product_prod(A,B)),set(product_prod(A,B))),F3))
       => ( ! [A6: A,B5: B] :
              ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),aa(set(product_prod(A,B)),set(product_prod(A,B)),F3,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F3)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P))))))
             => pp(aa(B,bool,aa(A,fun(B,bool),P,A6),B5)) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) ) ) ) ).

% lfp_induct2
tff(fact_5628_product__swap,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B4: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),product_Sigma(B,A,A5,aTP_Lamp_mi(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),A5)) ).

% product_swap
tff(fact_5629_prod_Oswap__def,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P3) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(product_prod(A,B),B,product_snd(A,B),P3)),aa(product_prod(A,B),A,product_fst(A,B),P3)) ).

% prod.swap_def
tff(fact_5630_ord_OatLeastAtMost__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),L: A,U: A] : set_atLeastAtMost(A,Less_eq,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_atLeast(A,Less_eq,L)),set_atMost(A,Less_eq,U)) ).

% ord.atLeastAtMost_def
tff(fact_5631_ord_OgreaterThanAtMost__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),L: A,U: A] : set_gr3752724095348155675AtMost(A,Less_eq,Less,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_greaterThan(A,Less,L)),set_atMost(A,Less_eq,U)) ).

% ord.greaterThanAtMost_def
tff(fact_5632_subset__mset_OgreaterThanAtMost__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_greaterThan(multiset(A),subset_mset(A),L)),set_atMost(multiset(A),subseteq_mset(A),U)) ).

% subset_mset.greaterThanAtMost_def
tff(fact_5633_subset__mset_OIcc__subset__Ici__iff,axiom,
    ! [A: $tType,L: multiset(A),H: multiset(A),L3: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),L,H)),set_atLeast(multiset(A),subseteq_mset(A),L3)))
    <=> ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),L),H))
        | pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),L3),L)) ) ) ).

% subset_mset.Icc_subset_Ici_iff
tff(fact_5634_ord_OgreaterThan__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),L: A] : set_greaterThan(A,Less,L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),Less,L)) ).

% ord.greaterThan_def
tff(fact_5635_ord_OatLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),L: A] : set_atLeast(A,Less_eq,L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),Less_eq,L)) ).

% ord.atLeast_def
tff(fact_5636_subset__mset_OatLeast__def,axiom,
    ! [A: $tType,L: multiset(A)] : set_atLeast(multiset(A),subseteq_mset(A),L) = aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),L)) ).

% subset_mset.atLeast_def
tff(fact_5637_subset__mset_OgreaterThan__def,axiom,
    ! [A: $tType,L: multiset(A)] : set_greaterThan(multiset(A),subset_mset(A),L) = aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),subset_mset(A),L)) ).

% subset_mset.greaterThan_def
tff(fact_5638_subset__mset_OIoi__le__Ico,axiom,
    ! [A: $tType,A3: multiset(A)] : pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),set_greaterThan(multiset(A),subset_mset(A),A3)),set_atLeast(multiset(A),subseteq_mset(A),A3))) ).

% subset_mset.Ioi_le_Ico
tff(fact_5639_subset__mset_Onot__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType,L: multiset(A)] : bot_bot(set(multiset(A))) != set_atLeast(multiset(A),subseteq_mset(A),L) ).

% subset_mset.not_empty_eq_Ici_eq_empty
tff(fact_5640_subset__mset_OatLeastAtMost__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_atLeastAtMost(multiset(A),subseteq_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_atLeast(multiset(A),subseteq_mset(A),L)),set_atMost(multiset(A),subseteq_mset(A),U)) ).

% subset_mset.atLeastAtMost_def
tff(fact_5641_subset__mset_OgreaterThanLessThan__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_gr287244882034783167ssThan(multiset(A),subset_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_greaterThan(multiset(A),subset_mset(A),L)),set_lessThan(multiset(A),subset_mset(A),U)) ).

% subset_mset.greaterThanLessThan_def
tff(fact_5642_subset__mset_OgreaterThanLessThan__eq,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_gr287244882034783167ssThan(multiset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_greaterThan(multiset(A),subset_mset(A),A3)),set_lessThan(multiset(A),subset_mset(A),B2)) ).

% subset_mset.greaterThanLessThan_eq
tff(fact_5643_ord_OgreaterThanLessThan__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),L: A,U: A] : set_gr287244882034783167ssThan(A,Less,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_greaterThan(A,Less,L)),set_lessThan(A,Less,U)) ).

% ord.greaterThanLessThan_def
tff(fact_5644_ord_OlessThan__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),U: A] : set_lessThan(A,Less,U) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less),U)) ).

% ord.lessThan_def
tff(fact_5645_subset__mset_OlessThan__def,axiom,
    ! [A: $tType,U: multiset(A)] : set_lessThan(multiset(A),subset_mset(A),U) = aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_acv(multiset(A),fun(multiset(A),bool)),U)) ).

% subset_mset.lessThan_def
tff(fact_5646_subset__mset_OIio__Int__singleton,axiom,
    ! [A: $tType,X: multiset(A),K: multiset(A)] :
      ( ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),X),K))
       => ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_lessThan(multiset(A),subset_mset(A),K)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A)))) ) )
      & ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),X),K))
       => ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_lessThan(multiset(A),subset_mset(A),K)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))) ) ) ) ).

% subset_mset.Iio_Int_singleton
tff(fact_5647_subset__mset_OatLeastLessThan__def,axiom,
    ! [A: $tType,L: multiset(A),U: multiset(A)] : set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),L,U) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_atLeast(multiset(A),subseteq_mset(A),L)),set_lessThan(multiset(A),subset_mset(A),U)) ).

% subset_mset.atLeastLessThan_def
tff(fact_5648_ord_OatLeastLessThan__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),L: A,U: A] : set_atLeastLessThan(A,Less_eq,Less,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_atLeast(A,Less_eq,L)),set_lessThan(A,Less,U)) ).

% ord.atLeastLessThan_def
tff(fact_5649_ord_OgreaterThanLessThan__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),A3: A,B2: A] : set_gr287244882034783167ssThan(A,Less,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_greaterThan(A,Less,A3)),set_lessThan(A,Less,B2)) ).

% ord.greaterThanLessThan_eq
tff(fact_5650_subset__mset_Olexordp__def,axiom,
    ! [A: $tType] : lexordp2(multiset(A),subset_mset(A)) = complete_lattice_lfp(fun(list(multiset(A)),fun(list(multiset(A)),bool)),aTP_Lamp_acr(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool)))) ).

% subset_mset.lexordp_def
tff(fact_5651_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_zr(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).

% ord_class.lexordp_def
tff(fact_5652_map__of__concat,axiom,
    ! [B: $tType,A: $tType,Xss: list(list(product_prod(A,B)))] : map_of(A,B,concat(product_prod(A,B),Xss)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(list(product_prod(A,B)),fun(A,option(B)),aTP_Lamp_adf(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Xss),aTP_Lamp_bk(A,option(B))) ).

% map_of_concat
tff(fact_5653_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A),Y: A,Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
              & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2)) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_5654_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X3))
         => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ) ).

% lexordp_irreflexive
tff(fact_5655_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) ) ) ).

% lexordp.Cons
tff(fact_5656_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_5657_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us: list(A),Vs: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
         => ( ! [A6: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),A6))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),Vs)) ) ) ) ).

% lexordp_append_leftD
tff(fact_5658_foldr__snd__zip,axiom,
    ! [B: $tType,A: $tType,C: $tType,Ys2: list(A),Xs: list(B),F3: fun(A,fun(C,C)),B2: C] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(B),nat,size_size(list(B)),Xs)))
     => ( aa(C,C,foldr(product_prod(B,A),C,aa(fun(B,fun(A,fun(C,C))),fun(product_prod(B,A),fun(C,C)),product_case_prod(B,A,fun(C,C)),aTP_Lamp_adg(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F3)),zip(B,A,Xs,Ys2)),B2) = aa(C,C,foldr(A,C,F3,Ys2),B2) ) ) ).

% foldr_snd_zip
tff(fact_5659_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : foldr(B,A,F3,aa(list(B),list(B),filter2(B,P),Xs)) = foldr(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_adh(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F3),P),Xs) ).

% foldr_filter
tff(fact_5660_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B),A3: A] : aa(A,A,foldr(B,A,F3,Xs),A3) = aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,aTP_Lamp_adi(fun(B,fun(A,A)),fun(A,fun(B,A)),F3)),A3),rev(B,Xs)) ).

% foldr_conv_foldl
tff(fact_5661_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),A3),Xs) = aa(A,A,foldr(B,A,aTP_Lamp_adj(fun(A,fun(B,A)),fun(B,fun(A,A)),F3),rev(B,Xs)),A3) ).

% foldl_conv_foldr
tff(fact_5662_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
         => ( ( ( A1 = nil(A) )
             => ! [Y3: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
           => ( ! [X3: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
                 => ! [Y3: A] :
                      ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3)) ) )
             => ~ ! [X3: A,Y3: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                       => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X3))
                           => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys3)) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_5663_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
        <=> ( ? [Y5: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) ) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs3),Ys4)) ) ) ) ) ).

% lexordp.simps
tff(fact_5664_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
         => ( ( ( Xs = nil(A) )
             => ! [Y3: A,Ys5: list(A)] : Ys2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
           => ( ! [X3: A] :
                  ( ? [Xs5: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs5)
                 => ! [Y3: A] :
                      ( ? [Ys5: list(A)] : Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3)) ) )
             => ~ ! [X3: A,Xs5: list(A)] :
                    ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs5) )
                   => ! [Ys5: list(A)] :
                        ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys5) )
                       => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs5),Ys5)) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_5665_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A),P: fun(list(A),fun(list(A),bool))] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
         => ( ! [Y3: A,Ys3: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))
           => ( ! [X3: A,Xs2: list(A),Y3: A,Ys3: list(A)] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))) )
             => ( ! [X3: A,Xs2: list(A),Ys3: list(A)] :
                    ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys3))
                   => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),Ys3))
                     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3))) ) )
               => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs),Ys2)) ) ) ) ) ) ).

% lexordp_induct
tff(fact_5666_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Us: list(A),Xs: list(A),Ys2: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),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),X),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))) ) ) ).

% lexordp_append_left_rightI
tff(fact_5667_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
        <=> ( ? [X4: A,Vs3: list(A)] : Ys2 = 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),X4),Vs3))
            | ? [Us3: list(A),A8: A,B13: A,Vs3: list(A),Ws: list(A)] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A8),B13))
                & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A8),Vs3)) )
                & ( Ys2 = 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),B13),Ws)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_5668_comp__fun__commute_Ofoldr__conv__foldl,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),Xs: list(A),A3: B] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(B,B,foldr(A,B,F3,Xs),A3) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_xk(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),A3),Xs) ) ) ).

% comp_fun_commute.foldr_conv_foldl
tff(fact_5669_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
        <=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))))) ) ) ).

% lexordp_conv_lexord
tff(fact_5670_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_adk(fun(B,A),fun(A,fun(B,fun(A,A))),F3),A3),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_5671_ord_Olexordp__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_zq(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less)) ).

% ord.lexordp_def
tff(fact_5672_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( ( ( Xs = nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y) = Y ) )
            & ( ( Xs != nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y) ) ) ) ) ) ).

% foldr_max_sorted
tff(fact_5673_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_adl(fun(A,set(B)),fun(A,filter(B)),F3)),X6)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X6))) ) ) ).

% INF_principal_finite
tff(fact_5674_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_se(A,fun(A,bool))) = ord_min(A) ) ) ).

% dual_max
tff(fact_5675_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_5676_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A3) = A3 ).

% max_nat.left_neutral
tff(fact_5677_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_5678_max__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),zero_zero(nat)) = A3 ).

% max_nat.right_neutral
tff(fact_5679_max__0L,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N2) = N2 ).

% max_0L
tff(fact_5680_max__0R,axiom,
    ! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N2),zero_zero(nat)) = N2 ).

% max_0R
tff(fact_5681_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N2)) ).

% max_Suc_Suc
tff(fact_5682_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).

% max.right_idem
tff(fact_5683_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).

% max.left_idem
tff(fact_5684_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_5685_principal__inject,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( principal(A,A5) = principal(A,B4) )
    <=> ( A5 = B4 ) ) ).

% principal_inject
tff(fact_5686_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% max.bounded_iff
tff(fact_5687_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_5688_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb1
tff(fact_5689_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb3
tff(fact_5690_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_5691_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2)) ) ) ) ).

% max_less_iff_conj
tff(fact_5692_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).

% max_bot2
tff(fact_5693_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).

% max_bot
tff(fact_5694_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).

% max_top
tff(fact_5695_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_5696_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = X ) ).

% max_min_same(1)
tff(fact_5697_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),X) = X ) ).

% max_min_same(2)
tff(fact_5698_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_5699_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = Y ) ).

% max_min_same(4)
tff(fact_5700_sup__principal,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),principal(A,A5)),principal(A,B4)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) ).

% sup_principal
tff(fact_5701_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_5702_principal__le__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),principal(A,A5)),principal(A,B4)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ).

% principal_le_iff
tff(fact_5703_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : aa(nat,nat,foldr(A,nat,aTP_Lamp_sw(A,fun(nat,nat)),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_5704_inf__principal,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),principal(A,A5)),principal(A,B4)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ).

% inf_principal
tff(fact_5705_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_5706_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_5707_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_5708_Int__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMost
tff(fact_5709_Int__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastLessThan
tff(fact_5710_SUP__principal,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),I5: 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_adm(fun(B,set(A)),fun(B,filter(A)),A5)),I5)) = principal(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))) ).

% SUP_principal
tff(fact_5711_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ).

% Max_insert
tff(fact_5712_max__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),M: A,N2: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F3,M)),aa(A,B,F3,N2)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_max(A),M),N2)) ) ) ) ).

% max_of_mono
tff(fact_5713_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.strict_coboundedI2
tff(fact_5714_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.strict_coboundedI1
tff(fact_5715_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_5716_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).

% max.strict_boundedE
tff(fact_5717_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y)) ) ) ) ).

% less_max_iff_disj
tff(fact_5718_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa3) = Xa3 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa3) = X5 ) ) ) ) ).

% max_def_raw
tff(fact_5719_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ) ).

% max_def
tff(fact_5720_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_5721_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_5722_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,D3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ) ).

% max.mono
tff(fact_5723_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.orderE
tff(fact_5724_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% max.orderI
tff(fact_5725_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).

% max.boundedE
tff(fact_5726_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)) ) ) ) ).

% max.boundedI
tff(fact_5727_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.order_iff
tff(fact_5728_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded1
tff(fact_5729_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded2
tff(fact_5730_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) ) ) ) ).

% le_max_iff_disj
tff(fact_5731_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_5732_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_5733_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI1
tff(fact_5734_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI2
tff(fact_5735_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ).

% Max.in_idem
tff(fact_5736_top__eq__principal__UNIV,axiom,
    ! [A: $tType] : top_top(filter(A)) = principal(A,top_top(set(A))) ).

% top_eq_principal_UNIV
tff(fact_5737_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.left_commute
tff(fact_5738_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3) ) ).

% max.commute
tff(fact_5739_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.assoc
tff(fact_5740_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_5741_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = B2 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = A3 ) ) ) ).

% ord.max_def
tff(fact_5742_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N2)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)) ).

% nat_mult_max_left
tff(fact_5743_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N2),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).

% nat_mult_max_right
tff(fact_5744_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N2),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ).

% nat_add_max_right
tff(fact_5745_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N2)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),Q3)) ).

% nat_add_max_left
tff(fact_5746_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_5747_complete__linorder__sup__max,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ( sup_sup(A) = ord_max(A) ) ) ).

% complete_linorder_sup_max
tff(fact_5748_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_5749_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C2)) ) ).

% min_max_distrib2
tff(fact_5750_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),A3)) ) ).

% min_max_distrib1
tff(fact_5751_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) ) ).

% max_min_distrib2
tff(fact_5752_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),A3)) ) ).

% max_min_distrib1
tff(fact_5753_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N2),M) ).

% nat_minus_add_max
tff(fact_5754_lessThan__Int__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_greaterThan(A),B2)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).

% lessThan_Int_lessThan
tff(fact_5755_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_adn(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_5756_Misc_Ofoldr__Cons,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),foldr(A,list(A),cons(A),Xs),nil(A)) = Xs ).

% Misc.foldr_Cons
tff(fact_5757_bot__eq__principal__empty,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).

% bot_eq_principal_empty
tff(fact_5758_principal__eq__bot__iff,axiom,
    ! [A: $tType,X6: set(A)] :
      ( ( principal(A,X6) = bot_bot(filter(A)) )
    <=> ( X6 = bot_bot(set(A)) ) ) ).

% principal_eq_bot_iff
tff(fact_5759_max__Suc1,axiom,
    ! [N2: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N2)),M) = case_nat(nat,aa(nat,nat,suc,N2),aTP_Lamp_ado(nat,fun(nat,nat),N2),M) ).

% max_Suc1
tff(fact_5760_max__Suc2,axiom,
    ! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N2)) = case_nat(nat,aa(nat,nat,suc,N2),aTP_Lamp_adp(nat,fun(nat,nat),N2),M) ).

% max_Suc2
tff(fact_5761_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ca(nat,fun(nat,bool)),aTP_Lamp_cb(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_5762_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_adq(list(B),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_adr(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_ads(list(B),bool)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_5763_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_se(A,fun(A,bool)),aTP_Lamp_adt(A,fun(A,bool))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_5764_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_adn(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_si(list(A),bool)),Xs)) ).

% transpose_max_length
tff(fact_5765_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_5766_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_5767_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_5768_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_5769_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( ( S = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = X ) )
            & ( ( S != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S)) ) ) ) ) ) ).

% Sup_insert_finite
tff(fact_5770_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_5771_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_5772_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,finite_finite2(A),N7))
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798349783984er_Max(A),N7)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,H),N7)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_5773_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B4)),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ) ).

% Max.subset
tff(fact_5774_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_5775_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5)),A5)) ) ) ) ) ).

% Max.closed
tff(fact_5776_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B4)) ) ) ) ) ) ) ).

% Max.union
tff(fact_5777_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = finite_fold(A,A,ord_max(A),X,A5) ) ) ) ).

% Max.eq_fold
tff(fact_5778_filter__conv__foldr,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = aa(list(A),list(A),foldr(A,list(A),aTP_Lamp_adu(fun(A,bool),fun(A,fun(list(A),list(A))),P),Xs),nil(A)) ).

% filter_conv_foldr
tff(fact_5779_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A3: nat] : aa(nat,nat,foldr(A,nat,aTP_Lamp_sw(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_5780_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Max.insert_remove
tff(fact_5781_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Max.remove
tff(fact_5782_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_adv(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Max.eq_fold'
tff(fact_5783_map__add__map__of__foldr,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,M,map_of(A,B,Ps)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(product_prod(A,B),fun(A,option(B)),aa(fun(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),product_case_prod(A,B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_adw(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps),M) ).

% map_add_map_of_foldr
tff(fact_5784_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_adx(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_5785_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_ady(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C2))) ) ).

% at_bot_sub
tff(fact_5786_finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( finite5375528669736107172at_top(A,A5) = principal(set(A),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),A5),bot_bot(set(set(A))))) ) ) ).

% finite_subsets_at_top_finite
tff(fact_5787_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_5788_finite__subsets__at__top__neq__bot,axiom,
    ! [A: $tType,A5: set(A)] : finite5375528669736107172at_top(A,A5) != bot_bot(filter(set(A))) ).

% finite_subsets_at_top_neq_bot
tff(fact_5789_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_5790_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A5: set(A)] : finite5375528669736107172at_top(A,A5) = 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_aea(set(A),fun(set(A),filter(set(A))),A5)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_aeb(set(A),fun(set(A),bool),A5)))) ).

% finite_subsets_at_top_def
tff(fact_5791_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aec(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Sup_fin.eq_fold'
tff(fact_5792_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aed(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Inf_fin.eq_fold'
tff(fact_5793_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aee(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Min.eq_fold'
tff(fact_5794_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Min_singleton
tff(fact_5795_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Sup_fin.singleton
tff(fact_5796_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Inf_fin.singleton
tff(fact_5797_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = A3 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_5798_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A3) = A3 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_5799_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_5800_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_5801_Min__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A5: set(B),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( A5 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_sb(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% Min_const
tff(fact_5802_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_5803_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_5804_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min_insert
tff(fact_5805_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_5806_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_5807_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),A3)) ) ) ) ).

% Min.coboundedI
tff(fact_5808_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ! [Y3: A] :
                ( pp(aa(set(A),bool,member(A,Y3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( pp(aa(set(A),bool,member(A,X),A5))
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_5809_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),X)) ) ) ) ).

% Min_le
tff(fact_5810_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5))) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_5811_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_5812_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_5813_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5)),A5)) ) ) ) ).

% Min_in
tff(fact_5814_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ).

% Min.in_idem
tff(fact_5815_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A3)) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_5816_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A5))) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_5817_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_5818_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_5819_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = M )
            <=> ( pp(aa(set(A),bool,member(A,M),A5))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X4)) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_5820_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Min_le_iff
tff(fact_5821_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A5) )
            <=> ( pp(aa(set(A),bool,member(A,M),A5))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X4)) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_5822_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A14)) ) ) ) ) ) ).

% Min.boundedE
tff(fact_5823_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( pp(aa(set(A),bool,member(A,A6),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A6)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5))) ) ) ) ) ).

% Min.boundedI
tff(fact_5824_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Min_less_iff
tff(fact_5825_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ! [B5: A] :
                ( pp(aa(set(A),bool,member(A,B5),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B5)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = A3 ) ) ) ) ).

% Min_insert2
tff(fact_5826_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).

% Min_Inf
tff(fact_5827_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = aa(set(A),A,lattic643756798350308766er_Min(A),X6) ) ) ) ) ).

% cInf_eq_Min
tff(fact_5828_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_5829_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A14)) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_5830_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( pp(aa(set(A),bool,member(A,A6),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A6)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_5831_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),X))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A14),X)) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_5832_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A6: A] :
                  ( pp(aa(set(A),bool,member(A,A6),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),X)) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_5833_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_5834_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_5835_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = aa(set(A),A,lattic5882676163264333800up_fin(A),X6) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_5836_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_5837_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_5838_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X6) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_5839_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_5840_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_5841_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N7: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M6),N7))
         => ( ( M6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N7))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N7)),aa(set(A),A,lattic643756798350308766er_Min(A),M6))) ) ) ) ) ).

% Min_antimono
tff(fact_5842_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B4)),aa(set(A),A,lattic643756798350308766er_Min(A),A5))) ) ) ) ) ).

% Min.subset_imp
tff(fact_5843_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,finite_finite2(A),N7))
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798350308766er_Min(A),N7)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,H),N7)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_5844_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B4)),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ) ).

% Min.subset
tff(fact_5845_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_5846_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5)),A5)) ) ) ) ) ).

% Min.closed
tff(fact_5847_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F3),A5)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_5848_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),aa(set(A),A,lattic643756798350308766er_Min(A),B4)) ) ) ) ) ) ) ).

% Min.union
tff(fact_5849_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = finite_fold(A,A,ord_min(A),X,A5) ) ) ) ).

% Min.eq_fold
tff(fact_5850_Min__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S: set(B),F3: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( S != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,A)),F3),K)),S)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,F3),S))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_5851_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_5852_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B4))) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_5853_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,finite_finite2(A),N7))
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic7752659483105999362nf_fin(A),N7)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image2(A,A,H),N7)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_5854_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,finite_finite2(A),N7))
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic5882676163264333800up_fin(A),N7)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image2(A,A,H),N7)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_5855_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_5856_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B4)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_5857_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A5)) ) ) ) ) ).

% Inf_fin.closed
tff(fact_5858_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_5859_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),A5)) ) ) ) ) ).

% Sup_fin.closed
tff(fact_5860_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_5861_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B4)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_5862_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B4)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_5863_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = finite_fold(A,A,inf_inf(A),X,A5) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_5864_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = finite_fold(A,A,sup_sup(A),X,A5) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_5865_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aef(set(A),fun(set(A),fun(A,bool)),A5),B4))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_5866_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aeg(set(A),fun(A,fun(A,bool)),A5),X))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_5867_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aeh(set(A),fun(set(A),fun(A,bool)),A5),B4))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_5868_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aei(set(A),fun(A,fun(A,bool)),A5),X))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_5869_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Min.insert_remove
tff(fact_5870_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Min.remove
tff(fact_5871_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Inf_fin.remove
tff(fact_5872_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_5873_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Sup_fin.remove
tff(fact_5874_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_5875_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
                & pp(aa(A,bool,P,X5)) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aej(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_5876_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_5877_card__Min__le__sum,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image2(A,nat,F3),A5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5))) ) ).

% card_Min_le_sum
tff(fact_5878_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_se(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_5879_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_se(A,fun(A,bool))) = ord_max(A) ) ) ).

% dual_min
tff(fact_5880_image__o__collect,axiom,
    ! [B: $tType,C: $tType,A: $tType,G3: fun(C,B),F4: set(fun(A,set(C)))] : bNF_collect(A,B,aa(set(fun(A,set(C))),set(fun(A,set(B))),image2(fun(A,set(C)),fun(A,set(B)),comp(set(C),set(B),A,image2(C,B,G3))),F4)) = aa(fun(A,set(C)),fun(A,set(B)),comp(set(C),set(B),A,image2(C,B,G3)),bNF_collect(A,C,F4)) ).

% image_o_collect
tff(fact_5881_subset__mset_Omin__arg__le_I2_J,axiom,
    ! [A: $tType,M: multiset(A),N2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2)))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2) = M ) ) ).

% subset_mset.min_arg_le(2)
tff(fact_5882_subset__mset_Omin__arg__le_I1_J,axiom,
    ! [A: $tType,N2: multiset(A),M: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),N2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2)))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2) = N2 ) ) ).

% subset_mset.min_arg_le(1)
tff(fact_5883_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = A3 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = B2 ) ) ) ).

% ord.min_def
tff(fact_5884_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_5885_linorder_OMax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).

% linorder.Max.cong
tff(fact_5886_collect__def,axiom,
    ! [A: $tType,B: $tType,F4: set(fun(B,set(A))),X: B] : aa(B,set(A),bNF_collect(B,A,F4),X) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(B,set(A))),set(set(A)),image2(fun(B,set(A)),set(A),aTP_Lamp_aek(B,fun(fun(B,set(A)),set(A)),X)),F4)) ).

% collect_def
tff(fact_5887_collect__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: set(fun(C,set(B))),G3: fun(A,C)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,bNF_collect(C,B,F4)),G3) = bNF_collect(A,B,aa(set(fun(C,set(B))),set(fun(A,set(B))),image2(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_ael(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),G3)),F4)) ).

% collect_comp
tff(fact_5888_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),T6: list(product_prod(A,C)),K: A,X: C] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,T6),K) = aa(C,option(C),some(C),X) )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_aem(fun(A,B),fun(A,fun(C,product_prod(B,C))),F3))),T6)),aa(A,B,F3,K)) = aa(C,option(C),some(C),X) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_5889_total__on__singleton,axiom,
    ! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_5890_nth__rotate,axiom,
    ! [A: $tType,N2: nat,Xs: list(A),M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rotate(A,M,Xs)),N2) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_5891_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : inj_on(A,B,F3,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_5892_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_5893_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( rotate(A,N2,Xs) = Xs ) ) ).

% rotate_length01
tff(fact_5894_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_aen(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_5895_inj__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),top_top(set(product_prod(A,B))))
    <=> inj_on(A,C,F3,top_top(set(A))) ) ).

% inj_apfst
tff(fact_5896_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F3,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_5897_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C),A5: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),product_Sigma(A,B,A5,aTP_Lamp_xu(A,set(B))))
    <=> inj_on(A,C,F3,A5) ) ).

% inj_on_apfst
tff(fact_5898_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C),A5: set(B)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xt(set(B),fun(A,set(B)),A5)))
    <=> inj_on(B,C,F3,A5) ) ).

% inj_on_apsnd
tff(fact_5899_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: A,A5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5))
    <=> ( inj_on(A,B,F3,A5)
        & ~ pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))))) ) ) ).

% inj_on_insert
tff(fact_5900_inj__on__Un__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))
     => ( ( aa(set(A),set(B),image2(A,B,F3),A5) = aa(set(A),set(B),image2(A,B,F3),B4) )
      <=> ( A5 = B4 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_5901_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B4: set(A),A5: set(A)] :
      ( inj_on(A,B,F3,B4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),aa(set(A),set(B),image2(A,B,F3),B4))) ) ) ).

% inj_on_strict_subset
tff(fact_5902_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4))
         => ( ( aa(set(A),set(B),image2(A,B,F3),A5) = aa(set(A),set(B),image2(A,B,F3),B4) )
          <=> ( A5 = B4 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_5903_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B4: set(A),A3: A,A5: set(A)] :
      ( inj_on(A,B,F3,B4)
     => ( pp(aa(set(A),bool,member(A,A3),B4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => ( pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),aa(set(A),set(B),image2(A,B,F3),A5)))
          <=> pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_5904_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(B,A),T2: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(B),set(A),image2(B,A,F3),T2)))
    <=> ? [U6: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U6),T2))
          & inj_on(B,A,F3,U6)
          & ( S = aa(set(B),set(A),image2(B,A,F3),U6) ) ) ) ).

% subset_image_inj
tff(fact_5905_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_rm(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_5906_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_aeo(fun(A,B),fun(A,fun(C,B)),F3),top_top(set(A))) ) ).

% inj_fun
tff(fact_5907_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(option(A),option(B),map_option(A,B,F3),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_5908_inj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N2: nat] :
      ( inj_on(A,A,F3,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),top_top(set(A))) ) ).

% inj_fn
tff(fact_5909_inj__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => inj_on(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ).

% inj_of_nat
tff(fact_5910_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
             => ( aa(A,B,F3,X3) != aa(A,B,F3,Y3) ) )
         => inj_on(A,B,F3,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_5911_inj__on__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(A)] :
      ( ( inj_on(A,B,F3,A5)
        | inj_on(A,B,F3,B4) )
     => inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ) ).

% inj_on_Int
tff(fact_5912_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B4: set(A),A5: set(A)] :
      ( inj_on(A,B,F3,B4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
       => inj_on(A,B,F3,A5) ) ) ).

% subset_inj_on
tff(fact_5913_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => inj_on(A,B,F3,B4) ) ) ).

% inj_on_subset
tff(fact_5914_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
             => ( pp(aa(set(A),bool,member(A,X3),A5))
               => ( pp(aa(set(A),bool,member(A,Y3),A5))
                 => ( aa(A,B,F3,X3) != aa(A,B,F3,Y3) ) ) ) )
         => ( ! [X3: A,Y3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => ( pp(aa(set(A),bool,member(A,Y3),A5))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3)) ) ) )
           => inj_on(A,B,F3,A5) ) ) ) ).

% linorder_inj_onI
tff(fact_5915_total__on__lex__prod,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R3: set(product_prod(A,A)),B4: set(B),S2: set(product_prod(B,B))] :
      ( total_on(A,A5,R3)
     => ( total_on(B,B4,S2)
       => total_on(product_prod(A,B),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),lex_prod(A,B,R3,S2)) ) ) ).

% total_on_lex_prod
tff(fact_5916_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,A5: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A5) ) ) ).

% inj_on_mult
tff(fact_5917_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(B,A),D4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( inj_on(B,A,F3,D4)
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_aep(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),A5),F3),D4)))) ) ) ).

% finite_inverse_image_gen
tff(fact_5918_inj__Some,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,option(A),some(A),A5) ).

% inj_Some
tff(fact_5919_inj__Suc,axiom,
    ! [N7: set(nat)] : inj_on(nat,nat,suc,N7) ).

% inj_Suc
tff(fact_5920_inj__on__id2,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,A,aTP_Lamp_cq(A,A),A5) ).

% inj_on_id2
tff(fact_5921_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A5: set(A)] : inj_on(A,A,aTP_Lamp_aeq(A,fun(A,A),A3),A5) ) ).

% inj_on_add'
tff(fact_5922_total__on__def,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( total_on(A,A5,R3)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),A5))
             => ( ( X4 != Xa )
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R3))
                  | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X4)),R3)) ) ) ) ) ) ).

% total_on_def
tff(fact_5923_total__onI,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => ( pp(aa(set(A),bool,member(A,Y3),A5))
           => ( ( X3 != Y3 )
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
                | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),R3)) ) ) ) )
     => total_on(A,A5,R3) ) ).

% total_onI
tff(fact_5924_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_wh(fun(A,B),fun(A,product_prod(A,B)),F3),X6) ).

% inj_on_convol_ident
tff(fact_5925_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_wh(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).

% inj_Pair(1)
tff(fact_5926_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_aer(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).

% inj_Pair(2)
tff(fact_5927_map__prod__inj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(A,B),A5: set(A),G3: fun(C,D),B4: set(C)] :
      ( inj_on(A,B,F3,A5)
     => ( inj_on(C,D,G3,B4)
       => inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F3,G3),product_Sigma(A,C,A5,aTP_Lamp_yh(set(C),fun(A,set(C)),B4))) ) ) ).

% map_prod_inj_on
tff(fact_5928_inj__swap,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_swap(A,B),A5) ).

% inj_swap
tff(fact_5929_total__on__empty,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R3) ).

% total_on_empty
tff(fact_5930_inj__on__Inter,axiom,
    ! [B: $tType,A: $tType,S: set(set(A)),F3: fun(A,B)] :
      ( ( S != bot_bot(set(set(A))) )
     => ( ! [A11: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),A11),S))
           => inj_on(A,B,F3,A11) )
       => inj_on(A,B,F3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ) ).

% inj_on_Inter
tff(fact_5931_inj__on__fst__map__to__set,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),map_to_set(A,B,M)) ).

% inj_on_fst_map_to_set
tff(fact_5932_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_cz(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_5933_inj__singleton,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,set(A),aTP_Lamp_pz(A,set(A)),A5) ).

% inj_singleton
tff(fact_5934_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( inj_on(B,A,F3,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_aes(set(A),fun(fun(B,A),fun(B,bool)),S),F3)))) ) ) ).

% finite_Collect
tff(fact_5935_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( inj_on(B,A,F3,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_aes(set(A),fun(fun(B,A),fun(B,bool)),A5),F3)))) ) ) ).

% finite_inverse_image
tff(fact_5936_sum_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A5: set(B)] :
          ( inj_on(B,A,G3,A5)
         => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_vd(A,A)),aa(set(B),set(A),image2(B,A,G3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) ) ) ) ).

% sum.image_eq
tff(fact_5937_prod_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),A5: set(B)] :
          ( inj_on(B,A,G3,A5)
         => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_aet(A,A)),aa(set(B),set(A),image2(B,A,G3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) ) ) ) ).

% prod.image_eq
tff(fact_5938_inj__on__diff__nat,axiom,
    ! [N7: set(nat),K: nat] :
      ( ! [N5: nat] :
          ( pp(aa(set(nat),bool,member(nat,N5),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N5)) )
     => inj_on(nat,nat,aTP_Lamp_qm(nat,fun(nat,nat),K),N7) ) ).

% inj_on_diff_nat
tff(fact_5939_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A5: 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_wx(A,fun(B,product_prod(B,A)))),A5) ).

% swap_inj_on
tff(fact_5940_inj__split__Cons,axiom,
    ! [A: $tType,X6: 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_uq(list(A),fun(A,list(A)))),X6) ).

% inj_split_Cons
tff(fact_5941_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),A15: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ? [F10: fun(A,B)] :
            ( inj_on(A,B,F10,A5)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F10),A5)),A15)) )
      <=> ? [G7: fun(B,A)] : aa(set(B),set(A),image2(B,A,G7),A15) = A5 ) ) ).

% inj_on_iff_surj
tff(fact_5942_finite__surj__inj,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),image2(A,A,F3),A5)))
       => inj_on(A,A,F3,A5) ) ) ).

% finite_surj_inj
tff(fact_5943_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B4))
       => ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ) ).

% inj_on_finite
tff(fact_5944_endo__inj__surj,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image2(A,A,F3),A5)),A5))
       => ( inj_on(A,A,F3,A5)
         => ( aa(set(A),set(A),image2(A,A,F3),A5) = A5 ) ) ) ) ).

% endo_inj_surj
tff(fact_5945_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),aa(set(A),set(B),image2(A,B,F3),B4)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ) ).

% inj_image_subset_iff
tff(fact_5946_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4))
         => ( aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) = 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,F3),A5)),aa(set(A),set(B),image2(A,B,F3),B4)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_5947_image__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) = 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,F3),A5)),aa(set(A),set(B),image2(A,B,F3),B4)) ) ) ).

% image_Int
tff(fact_5948_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)),C4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C4))
         => ( aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = 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,F3),A5)),aa(set(A),set(B),image2(A,B,F3),B4)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_5949_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(B),nat,finite_card(B),A5)))
     => ~ inj_on(B,A,F3,A5) ) ).

% pigeonhole
tff(fact_5950_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys2) )
     => ( inj_on(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys2)))
       => ( Xs = Ys2 ) ) ) ).

% map_inj_on
tff(fact_5951_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys2: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)))
     => ( ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys2) )
      <=> ( Xs = Ys2 ) ) ) ).

% inj_on_map_eq_map
tff(fact_5952_finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( inj_on(B,A,H,A5)
       => pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H),F4)),A5))) ) ) ).

% finite_vimage_IntI
tff(fact_5953_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_aev(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_5954_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),F3: fun(B,C)] :
      ( ! [I3: A,J2: A] :
          ( pp(aa(set(A),bool,member(A,I3),I5))
         => ( pp(aa(set(A),bool,member(A,J2),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A5,I3)),aa(A,set(B),A5,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A5,J2)),aa(A,set(B),A5,I3))) ) ) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => inj_on(B,C,F3,aa(A,set(B),A5,I3)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_5955_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),aTP_Lamp_aew(fun(A,B),fun(A,fun(A,bool)),F3),Y)),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_5956_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F3: fun(B,C),A5: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => inj_on(B,C,F3,aa(A,set(B),A5,I3)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5))) ) ) ).

% inj_on_INTER
tff(fact_5957_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ? [N5: nat,F2: fun(nat,A)] :
          ( ( A5 = aa(set(nat),set(A),image2(nat,A,F2),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N5))) )
          & inj_on(nat,A,F2,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N5))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_5958_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ~ ! [F2: fun(A,nat)] :
            ( ? [N5: nat] : aa(set(A),set(nat),image2(A,nat,F2),A5) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N5))
           => ~ inj_on(A,nat,F2,A5) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_5959_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ? [F2: fun(A,nat),N5: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F2),A5) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N5)) )
          & inj_on(A,nat,F2,A5) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_5960_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_5961_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S: set(A),T2: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(B),bool,finite_finite2(B),T2))
       => ( ( aa(set(A),nat,finite_card(A),S) = aa(set(B),nat,finite_card(B),T2) )
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),S)),T2))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),T2))
                 => ? [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),S))
                      & ( aa(A,B,F3,Xa) = X4 ) ) )
            <=> inj_on(A,B,F3,S) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_5962_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B4: set(B),G3: fun(B,A)] :
      ( inj_on(A,B,F3,A5)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B4))
       => ( inj_on(B,A,G3,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G3),B4)),A5))
           => ( pp(aa(set(A),bool,finite_finite2(A),A5))
             => ( pp(aa(set(B),bool,finite_finite2(B),B4))
               => ( aa(set(A),nat,finite_card(A),A5) = aa(set(B),nat,finite_card(B),B4) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_5963_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B4: set(B),A5: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),image2(A,B,F3),A5)))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)),A5)) ) ) ).

% vimage_subsetI
tff(fact_5964_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),uminus_uminus(set(A)),A5))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)))) ) ).

% inj_image_Compl_subset
tff(fact_5965_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X3: nat] :
            ( pp(aa(set(nat),bool,member(nat,X3),I5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs))) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_5966_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
    <=> ? [F10: fun(nat,A)] :
          ( inj_on(nat,A,F10,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F10),top_top(set(nat)))),S)) ) ) ).

% infinite_iff_countable_subset
tff(fact_5967_infinite__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
     => ? [F2: fun(nat,A)] :
          ( inj_on(nat,A,F2,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat)))),S)) ) ) ).

% infinite_countable_subset
tff(fact_5968_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),G3: fun(A,B),B4: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( inj_on(A,B,G3,B4)
       => ( ( 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,F3),A5)),aa(set(A),set(B),image2(A,B,G3),B4)) = 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_aex(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F3),A5),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_5969_set__to__map__simp,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B)),K: 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),K) = aa(B,option(B),some(B),V2) )
      <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),S)) ) ) ).

% set_to_map_simp
tff(fact_5970_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),C4: set(A),A5: set(C),B4: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F3,C4)
     => ( ! [X3: C] :
            ( pp(aa(set(C),bool,member(C,X3),A5))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(C,set(A),B4,X3)),C4)) )
       => ( pp(aa(set(C),bool,member(C,J),A5))
         => ( aa(set(A),set(B),image2(A,B,F3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B4),A5))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_aey(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F3),B4)),A5)) ) ) ) ) ).

% image_INT
tff(fact_5971_inj__on__funpow__least,axiom,
    ! [A: $tType,N2: nat,F3: fun(A,A),S2: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),S2) = S2 )
     => ( ! [M5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M5))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N2))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M5),F3),S2) != S2 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_aez(fun(A,A),fun(A,fun(nat,A)),F3),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)) ) ) ).

% inj_on_funpow_least
tff(fact_5972_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => ( ? [F10: fun(A,B)] :
              ( inj_on(A,B,F10,A5)
              & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F10),A5)),B4)) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B4))) ) ) ) ).

% inj_on_iff_card_le
tff(fact_5973_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B4: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B4))
       => ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B4))) ) ) ) ).

% card_inj_on_le
tff(fact_5974_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B4)))
         => ? [F2: fun(A,B)] :
              ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A5)),B4))
              & inj_on(A,B,F2,A5) ) ) ) ) ).

% card_le_inj
tff(fact_5975_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))
    <=> ( inj_on(A,B,F3,A5)
        & inj_on(A,B,F3,B4)
        & ( 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,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4))),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A5))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_5976_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),aa(set(A),set(B),image2(A,B,F3),top_top(set(A)))))
       => ( aa(set(A),nat,finite_card(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)) = aa(set(B),nat,finite_card(B),A5) ) ) ) ).

% card_vimage_inj
tff(fact_5977_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),D4: set(A),A5: set(B)] :
      ( inj_on(A,B,F3,D4)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)),D4))),aa(set(B),nat,finite_card(B),A5))) ) ) ).

% card_vimage_inj_on_le
tff(fact_5978_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),I5: set(B)] :
    ? [F2: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5)))
      & pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),aa(set(A),set(product_prod(B,A)),image2(A,product_prod(B,A),F2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5)))),product_Sigma(B,A,I5,A5))) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_5979_map__sorted__distinct__set__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Ys2: list(B)] :
          ( inj_on(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys2)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
           => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
             => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Ys2))
               => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Ys2))
                 => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys2) )
                   => ( Xs = Ys2 ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_5980_sort__key__inj__key__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),Ys2: list(B),F3: fun(B,A)] :
          ( ( mset(B,Xs) = mset(B,Ys2) )
         => ( inj_on(B,A,F3,aa(list(B),set(B),set2(B),Xs))
           => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Ys2))
             => ( aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = Ys2 ) ) ) ) ) ).

% sort_key_inj_key_eq
tff(fact_5981_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F3: fun(A,B),G3: fun(C,B),A5: set(A),B4: set(C)] :
          ( inj_on(product_prod(A,C),B,aa(fun(A,fun(C,B)),fun(product_prod(A,C),B),product_case_prod(A,C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_dv(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F3),G3)),product_Sigma(A,C,A5,aTP_Lamp_yh(set(C),fun(A,set(C)),B4)))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),B4)) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),id(B)),aa(fun(B,bool),set(B),collect(B),aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_afa(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F3),G3),A5),B4))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_5982_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),X: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_afb(fun(A,A),fun(A,fun(A,bool)),P3),X))))
       => ~ ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N5),P3),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_5983_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A5))
     => ( inj_on(A,B,F3,A5)
       => ( aa(list(B),list(A),map(B,A,inv_on(A,B,F3,A5)),aa(list(A),list(B),map(A,B,F3),L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_5984_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),A3: B] :
      ( inj_on(A,B,F3,A5)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_afc(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F3),A5),A3))),bot_bot(set(A))))) ) ).

% inj_on_vimage_singleton
tff(fact_5985_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: B] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the(A,aa(B,fun(A,bool),aTP_Lamp_afd(fun(A,B),fun(B,fun(A,bool)),F3),A3))),bot_bot(set(A))))) ) ).

% inj_vimage_singleton
tff(fact_5986_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_afe(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_5987_inv__on__f__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),X: A] :
      ( inj_on(A,B,F3,A5)
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( aa(B,A,inv_on(A,B,F3,A5),aa(A,B,F3,X)) = X ) ) ) ).

% inv_on_f_f
tff(fact_5988_f__inv__on__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),image2(B,A,F3),A5)))
     => ( aa(B,A,F3,aa(A,B,inv_on(B,A,F3,A5),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_5989_inv__on__f__range,axiom,
    ! [A: $tType,B: $tType,Y: A,F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),image2(B,A,F3),A5)))
     => pp(aa(set(B),bool,member(B,aa(A,B,inv_on(B,A,F3,A5),Y)),A5)) ) ).

% inv_on_f_range
tff(fact_5990_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P)) = the(product_prod(A,B),aTP_Lamp_aff(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P)) ).

% The_case_prod
tff(fact_5991_the__elem__def,axiom,
    ! [A: $tType,X6: set(A)] : the_elem(A,X6) = the(A,aTP_Lamp_afg(set(A),fun(A,bool),X6)) ).

% the_elem_def
tff(fact_5992_old_Orec__prod__def,axiom,
    ! [T: $tType,B: $tType,A: $tType,X5: fun(A,fun(B,T)),Xa3: product_prod(A,B)] : product_rec_prod(A,B,T,X5,Xa3) = the(T,product_rec_set_prod(A,B,T,X5,Xa3)) ).

% old.rec_prod_def
tff(fact_5993_old_Orec__nat__def,axiom,
    ! [T: $tType,X5: T,Xa3: fun(nat,fun(T,T)),Xb2: nat] : aa(nat,T,rec_nat(T,X5,Xa3),Xb2) = the(T,rec_set_nat(T,X5,Xa3,Xb2)) ).

% old.rec_nat_def
tff(fact_5994_the__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ( X3 = A3 ) )
       => ( the(A,P) = A3 ) ) ) ).

% the_equality
tff(fact_5995_the__eq__trivial,axiom,
    ! [A: $tType,A3: A] : the(A,aTP_Lamp_br(A,fun(A,bool),A3)) = A3 ).

% the_eq_trivial
tff(fact_5996_the__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : the(A,aa(A,fun(A,bool),fequal(A),X)) = X ).

% the_sym_eq_trivial
tff(fact_5997_the1__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X5 ) ) )
     => ( pp(aa(A,bool,P,A3))
       => ( the(A,P) = A3 ) ) ) ).

% the1_equality
tff(fact_5998_the1I2,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X5 ) ) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,the(A,P))) ) ) ).

% the1I2
tff(fact_5999_If__def,axiom,
    ! [A: $tType,P: bool,X: A,Y: A] :
      ( ( pp(P)
       => ( X = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_afh(bool,fun(A,fun(A,fun(A,bool))),P),X),Y)) ) )
      & ( ~ pp(P)
       => ( Y = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_afh(bool,fun(A,fun(A,fun(A,bool))),P),X),Y)) ) ) ) ).

% If_def
tff(fact_6000_theI2,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ( X3 = A3 ) )
       => ( ! [X3: A] :
              ( pp(aa(A,bool,P,X3))
             => pp(aa(A,bool,Q,X3)) )
         => pp(aa(A,bool,Q,the(A,P))) ) ) ) ).

% theI2
tff(fact_6001_theI_H,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X5 ) ) )
     => pp(aa(A,bool,P,the(A,P))) ) ).

% theI'
tff(fact_6002_theI,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ( X3 = A3 ) )
       => pp(aa(A,bool,P,the(A,P))) ) ) ).

% theI
tff(fact_6003_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_afi(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_6004_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),X: B,Y: A,Z2: A] :
      ( ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) )
     => ( inj_on(B,option(A),M,dom(B,A,M))
       => ( ~ pp(aa(set(A),bool,member(A,Z2),ran(B,A,M)))
         => ( ran(B,A,fun_upd(B,option(A),M,X,aa(A,option(A),some(A),Z2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),ran(B,A,M)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z2),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_6005_restrict__map__self,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : restrict_map(A,B,M,dom(A,B,M)) = M ).

% restrict_map_self
tff(fact_6006_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] :
      ( ( dom(A,B,F3) = bot_bot(set(A)) )
    <=> ! [X4: A] : aa(A,option(B),F3,X4) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_6007_dom__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),N2: fun(A,option(B))] : dom(A,B,map_add(A,B,M,N2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,N2)),dom(A,B,M)) ).

% dom_map_add
tff(fact_6008_dom__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),A5: set(A)] : dom(A,B,restrict_map(A,B,M,A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M)),A5) ).

% dom_restrict
tff(fact_6009_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_bk(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_6010_map__update__eta__repair_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V2: B,M: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_afj(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),K),V2),M)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),dom(A,B,M)) ).

% map_update_eta_repair(1)
tff(fact_6011_dom__const_H,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : dom(A,B,aTP_Lamp_afk(fun(A,B),fun(A,option(B)),F3)) = top_top(set(A)) ).

% dom_const'
tff(fact_6012_restrict__map__inv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X5: A] : aa(A,option(B),restrict_map(A,B,F3,aa(set(A),set(A),uminus_uminus(set(A)),dom(A,B,F3))),X5) = none(B) ).

% restrict_map_inv
tff(fact_6013_map__add__upd__left,axiom,
    ! [A: $tType,B: $tType,M: A,E22: fun(A,option(B)),E1: fun(A,option(B)),U1: B] :
      ( ~ pp(aa(set(A),bool,member(A,M),dom(A,B,E22)))
     => ( map_add(A,B,fun_upd(A,option(B),E1,M,aa(B,option(B),some(B),U1)),E22) = fun_upd(A,option(B),map_add(A,B,E1,E22),M,aa(B,option(B),some(B),U1)) ) ) ).

% map_add_upd_left
tff(fact_6014_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: option(B),F3: fun(A,option(B)),X: A] :
      ( ( ( Y = none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F3,X,Y)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) )
      & ( ( Y != none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F3,X,Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),dom(A,B,F3)) ) ) ) ).

% dom_fun_upd
tff(fact_6015_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xs: list(A),Ys2: list(B)] : dom(A,B,map_upds(A,B,M,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))),dom(A,B,M)) ).

% dom_map_upds
tff(fact_6016_ran__add,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F3)),dom(A,B,G3)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,F3,G3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F3)),ran(A,B,G3)) ) ) ).

% ran_add
tff(fact_6017_ran__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,M1,M22)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,M1)),ran(A,B,M22)) ) ) ).

% ran_map_add
tff(fact_6018_le__map__dom__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [M: fun(A,option(B)),M8: fun(A,option(B))] :
          ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M8))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),dom(A,B,M)),dom(A,B,M8))) ) ) ).

% le_map_dom_mono
tff(fact_6019_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),X),Y))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_6020_abs__rat__def,axiom,
    ! [A3: rat] :
      ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A3) = aa(rat,rat,uminus_uminus(rat),A3) ) )
      & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A3) = A3 ) ) ) ).

% abs_rat_def
tff(fact_6021_sgn__rat__def,axiom,
    ! [A3: rat] :
      ( ( ( A3 = zero_zero(rat) )
       => ( aa(rat,rat,sgn_sgn(rat),A3) = zero_zero(rat) ) )
      & ( ( A3 != zero_zero(rat) )
       => ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
           => ( aa(rat,rat,sgn_sgn(rat),A3) = one_one(rat) ) )
          & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
           => ( aa(rat,rat,sgn_sgn(rat),A3) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_6022_insert__dom,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),X: B,Y: A] :
      ( ( aa(B,option(A),F3,X) = aa(A,option(A),some(A),Y) )
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),dom(B,A,F3)) = dom(B,A,F3) ) ) ).

% insert_dom
tff(fact_6023_domI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M,A3) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(B),bool,member(B,A3),dom(B,A,M))) ) ).

% domI
tff(fact_6024_domD,axiom,
    ! [A: $tType,B: $tType,A3: A,M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,member(A,A3),dom(A,B,M)))
     => ? [B5: B] : aa(A,option(B),M,A3) = aa(B,option(B),some(B),B5) ) ).

% domD
tff(fact_6025_nempty__dom,axiom,
    ! [B: $tType,A: $tType,E3: fun(A,option(B))] :
      ( ~ ! [X5: A] : aa(A,option(B),E3,X5) = none(B)
     => ~ ! [M5: A] : ~ pp(aa(set(A),bool,member(A,M5),dom(A,B,E3))) ) ).

% nempty_dom
tff(fact_6026_dom__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_wa(fun(A,option(B)),fun(A,bool),M)) ).

% dom_def
tff(fact_6027_dom__map__option,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,fun(C,B)),M: fun(A,option(C))] : dom(A,B,aa(fun(A,option(C)),fun(A,option(B)),aTP_Lamp_afl(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),F3),M)) = dom(A,C,M) ).

% dom_map_option
tff(fact_6028_map__dom__ran__finite,axiom,
    ! [B: $tType,A: $tType,M6: fun(A,option(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M6)))
     => pp(aa(set(B),bool,finite_finite2(B),ran(A,B,M6))) ) ).

% map_dom_ran_finite
tff(fact_6029_dom__if,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F3: fun(A,option(B)),G3: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_afm(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),P),F3),G3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F3)),aa(fun(A,bool),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,G3)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ci(fun(A,bool),fun(A,bool),P)))) ).

% dom_if
tff(fact_6030_map__add__left__comm,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,option(B)),B4: fun(A,option(B)),C4: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,A5)),dom(A,B,B4)) = bot_bot(set(A)) )
     => ( map_add(A,B,A5,map_add(A,B,B4,C4)) = map_add(A,B,B4,map_add(A,B,A5,C4)) ) ) ).

% map_add_left_comm
tff(fact_6031_map__add__comm,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( map_add(A,B,M1,M22) = map_add(A,B,M22,M1) ) ) ).

% map_add_comm
tff(fact_6032_restrict__map__eq_I1_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A5: set(B),K: B] :
      ( ( aa(B,option(A),restrict_map(B,A,M,A5),K) = none(A) )
    <=> ~ pp(aa(set(B),bool,member(B,K),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,A,M)),A5))) ) ).

% restrict_map_eq(1)
tff(fact_6033_map__card__eq__iff,axiom,
    ! [B: $tType,A: $tType,M6: fun(A,option(B)),X: A,Y: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M6)))
     => ( ( aa(set(A),nat,finite_card(A),dom(A,B,M6)) = aa(set(B),nat,finite_card(B),ran(A,B,M6)) )
       => ( pp(aa(set(A),bool,member(A,X),dom(A,B,M6)))
         => ( ( aa(A,option(B),M6,X) = aa(A,option(B),M6,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% map_card_eq_iff
tff(fact_6034_finite__map__to__set,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),map_to_set(A,B,M)))
    <=> pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M))) ) ).

% finite_map_to_set
tff(fact_6035_map__to__set__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),map_to_set(A,B,M)) ).

% map_to_set_dom
tff(fact_6036_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_6037_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => pp(aa(set(fun(A,option(B))),bool,finite_finite2(fun(A,option(B))),aa(fun(fun(A,option(B)),bool),set(fun(A,option(B))),collect(fun(A,option(B))),aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_afn(set(A),fun(set(B),fun(fun(A,option(B)),bool)),A5),B4)))) ) ) ).

% finite_set_of_finite_maps
tff(fact_6038_card__map__to__set,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),map_to_set(A,B,M)) = aa(set(A),nat,finite_card(A),dom(A,B,M)) ).

% card_map_to_set
tff(fact_6039_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(fun(A,option(B)),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M)))
     => ( pp(aa(fun(A,option(B)),bool,P,aTP_Lamp_bk(A,option(B))))
       => ( ! [K2: A,V3: B,M5: fun(A,option(B))] :
              ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M5)))
             => ( ~ pp(aa(set(A),bool,member(A,K2),dom(A,B,M5)))
               => ( pp(aa(fun(A,option(B)),bool,P,M5))
                 => pp(aa(fun(A,option(B)),bool,P,fun_upd(A,option(B),M5,K2,aa(B,option(B),some(B),V3)))) ) ) )
         => pp(aa(fun(A,option(B)),bool,P,M)) ) ) ) ).

% finite_Map_induct
tff(fact_6040_ran__is__image,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,option(A))] : ran(B,A,M6) = aa(set(B),set(A),image2(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),M6)),dom(B,A,M6)) ).

% ran_is_image
tff(fact_6041_map__add__distinct__le,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [M: fun(A,option(B)),M8: fun(A,option(B)),N2: fun(A,option(B)),N6: fun(A,option(B))] :
          ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M8))
         => ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),N2),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)) )
             => pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N2)),map_add(A,B,M8,N6))) ) ) ) ) ).

% map_add_distinct_le
tff(fact_6042_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_afo(fun(A,option(B)),fun(A,product_prod(A,B)),M)),dom(A,B,M)) ).

% graph_eq_to_snd_dom
tff(fact_6043_graph__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( graph(A,B,map_add(A,B,M1,M22)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),graph(A,B,M1)),graph(A,B,M22)) ) ) ).

% graph_map_add
tff(fact_6044_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
    <=> ? [V5: B] : F3 = fun_upd(A,option(B),aTP_Lamp_bk(A,option(B)),X,aa(B,option(B),some(B),V5)) ) ).

% dom_eq_singleton_conv
tff(fact_6045_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),M: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_afo(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).

% map_of_map_keys
tff(fact_6046_inj__on__map__the,axiom,
    ! [B: $tType,A: $tType,D4: set(A),M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D4),dom(A,B,M)))
     => ( inj_on(A,option(B),M,D4)
       => inj_on(A,B,aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),M),D4) ) ) ).

% inj_on_map_the
tff(fact_6047_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),G3: fun(A,option(B)),A5: set(A)] : dom(A,B,override_on(A,option(B),F3,G3,A5)) = 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,F3)),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_afp(fun(A,option(B)),fun(set(A),fun(A,bool)),G3),A5)))),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_afq(fun(A,option(B)),fun(set(A),fun(A,bool)),G3),A5))) ).

% dom_override_on
tff(fact_6048_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q3),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).

% normalize_negative
tff(fact_6049_override__on__emptyset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G3: fun(A,B)] : override_on(A,B,F3,G3,bot_bot(set(A))) = F3 ).

% override_on_emptyset
tff(fact_6050_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_6051_normalize__denom__pos,axiom,
    ! [R3: product_prod(int,int),P3: int,Q3: int] :
      ( ( normalize(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q3)) ) ).

% normalize_denom_pos
tff(fact_6052_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P3: int,R3: int] :
      ( ( Q3 != zero_zero(int) )
     => ( ( S2 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R3),S2)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R3),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_6053_normalize__def,axiom,
    ! [P3: product_prod(int,int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)))
       => ( normalize(P3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)))
       => ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int) )
           => ( normalize(P3) = 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(product_prod(int,int),int,product_snd(int,int),P3) != zero_zero(int) )
           => ( normalize(P3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))) ) ) ) ) ) ).

% normalize_def
tff(fact_6054_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_6055_gcd__pos__int,axiom,
    ! [M: int,N2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N2)))
    <=> ( ( M != zero_zero(int) )
        | ( N2 != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_6056_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% Gcd_2
tff(fact_6057_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_6058_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_6059_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_6060_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_6061_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ).

% gcd_ge_0_int
tff(fact_6062_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,K: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))) ) ).

% gcd_dvd_prod
tff(fact_6063_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,N2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),N2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N2) ) ).

% gcd_add_mult
tff(fact_6064_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_6065_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_6066_quotient__of__div,axiom,
    ! [R3: rat,N2: int,D3: int] :
      ( ( quotient_of(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),N2),D3) )
     => ( R3 = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),N2)),aa(int,rat,ring_1_of_int(rat),D3)) ) ) ).

% quotient_of_div
tff(fact_6067_gcd__le1__int,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),A3)) ) ).

% gcd_le1_int
tff(fact_6068_gcd__le2__int,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),B2)) ) ).

% gcd_le2_int
tff(fact_6069_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun(int,bool)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y)))) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y))) ) )
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y)))) ) )
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ) ) ).

% gcd_cases_int
tff(fact_6070_gcd__unique__int,axiom,
    ! [D3: int,A3: int,B2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D3))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),A3))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),B2))
        & ! [E6: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E6),A3))
              & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E6),B2)) )
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E6),D3)) ) )
    <=> ( D3 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2) ) ) ).

% gcd_unique_int
tff(fact_6071_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,X,Y)) ) ) ).

% gcd_non_0_int
tff(fact_6072_quotient__of__denom__pos,axiom,
    ! [R3: rat,P3: int,Q3: int] :
      ( ( quotient_of(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q3)) ) ).

% quotient_of_denom_pos
tff(fact_6073_quotient__of__denom__pos_H,axiom,
    ! [R3: rat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R3)))) ).

% quotient_of_denom_pos'
tff(fact_6074_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N2) = aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_afr(int,fun(int,fun(int,bool)),N2),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_6075_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_afs(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_uminus_code
tff(fact_6076_rat__times__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_afu(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_times_code
tff(fact_6077_rat__divide__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_afw(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_divide_code
tff(fact_6078_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P3),Q3))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_afy(rat,fun(int,fun(int,bool)),Q3)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_6079_rat__less__eq__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),P3),Q3))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_aga(rat,fun(int,fun(int,bool)),Q3)),quotient_of(P3))) ) ).

% rat_less_eq_code
tff(fact_6080_rat__floor__code,axiom,
    ! [P3: rat] : archim6421214686448440834_floor(rat,P3) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),divide_divide(int)),quotient_of(P3)) ).

% rat_floor_code
tff(fact_6081_rat__abs__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_agb(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_abs_code
tff(fact_6082_rat__plus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_agd(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_plus_code
tff(fact_6083_rat__minus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_agf(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).

% rat_minus_code
tff(fact_6084_rat__sgn__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P3)))),one_one(int)) ).

% rat_sgn_code
tff(fact_6085_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_6086_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_agg(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_inverse_code
tff(fact_6087_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N2)))
    <=> ( ( M != zero_zero(nat) )
        | ( N2 != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_6088_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_6089_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_6090_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_6091_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_6092_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_6093_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_6094_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% inverse_positive_iff_positive
tff(fact_6095_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_6096_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_6097_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_6098_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_6099_gcd__le1__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),A3)) ) ).

% gcd_le1_nat
tff(fact_6100_gcd__le2__nat,axiom,
    ! [B2: nat,A3: nat] :
      ( ( B2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2)) ) ).

% gcd_le2_nat
tff(fact_6101_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N2)),N2) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N2) ) ) ).

% gcd_diff1_nat
tff(fact_6102_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)),N2) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N2) ) ) ).

% gcd_diff2_nat
tff(fact_6103_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa2))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa2))) ) ).

% mult_inverse_of_int_commute
tff(fact_6104_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A3) ) ).

% divide_inverse_commute
tff(fact_6105_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_6106_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_6107_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_inverse_distrib
tff(fact_6108_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_6109_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa2))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa2))) ) ).

% mult_inverse_of_nat_commute
tff(fact_6110_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_6111_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_6112_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = B2 ) ) ) ).

% inverse_unique
tff(fact_6113_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ).

% positive_imp_inverse_positive
tff(fact_6114_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_6115_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_6116_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_6117_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_6118_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_6119_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less
tff(fact_6120_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less
tff(fact_6121_Gcd__in,axiom,
    ! [A5: set(nat)] :
      ( ! [A6: nat,B5: nat] :
          ( pp(aa(set(nat),bool,member(nat,A6),A5))
         => ( pp(aa(set(nat),bool,member(nat,B5),A5))
           => pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A6),B5)),A5)) ) )
     => ( ( A5 != bot_bot(set(nat)) )
       => pp(aa(set(nat),bool,member(nat,gcd_Gcd(nat,A5)),A5)) ) ) ).

% Gcd_in
tff(fact_6122_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_6123_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_6124_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% le_imp_inverse_le
tff(fact_6125_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le
tff(fact_6126_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% inverse_le_1_iff
tff(fact_6127_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% one_less_inverse_iff
tff(fact_6128_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_less_inverse
tff(fact_6129_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_6130_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_6131_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_6132_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_6133_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A3: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) )
      | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_6134_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_6135_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_6136_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).

% one_le_inverse_iff
tff(fact_6137_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% inverse_less_1_iff
tff(fact_6138_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_le_inverse
tff(fact_6139_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N5)))),X)) ) ) ).

% reals_Archimedean
tff(fact_6140_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_im(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_6141_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N5))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_6142_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N2: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_6143_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N2) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_agh(nat,fun(nat,fun(nat,bool)),N2),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6144_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa2) = Y )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero(nat) )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa2),modulo_modulo(nat,X,Xa2)) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))) ) ) ) ).

% gcd_nat.pelims
tff(fact_6145_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_6146_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_6147_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_6148_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_6149_Frct__code__post_I8_J,axiom,
    ! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).

% Frct_code_post(8)
tff(fact_6150_Frct__code__post_I7_J,axiom,
    ! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),A3)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).

% Frct_code_post(7)
tff(fact_6151_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_6152_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_6153_rat__floor__lemma,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),one_one(int))))) ) ).

% rat_floor_lemma
tff(fact_6154_If__the__inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,I2: A,C4: set(A),G3: fun(A,B),X: A] :
      ( pp(aa(set(A),bool,member(A,I2),C4))
     => ( inj_on(A,B,G3,C4)
       => ( aa(A,A,aa(fun(A,B),fun(A,A),comp(B,A,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_agi(set(A),fun(fun(A,B),fun(A,fun(B,A))),C4),G3),X)),G3),I2) = aa(A,A,id(A),I2) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_6155_less__rat,axiom,
    ! [B2: int,D3: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)))) ) ) ) ).

% less_rat
tff(fact_6156_le__rat,axiom,
    ! [B2: int,D3: int,A3: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)))) ) ) ) ).

% le_rat
tff(fact_6157_quotient__of__eq,axiom,
    ! [A3: int,B2: int,P3: int,Q3: int] :
      ( ( quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P3),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).

% quotient_of_eq
tff(fact_6158_Rat__induct__pos,axiom,
    ! [P: fun(rat,bool),Q3: rat] :
      ( ! [A6: int,B5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
         => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A6),B5))) )
     => pp(aa(rat,bool,P,Q3)) ) ).

% Rat_induct_pos
tff(fact_6159_normalize__eq,axiom,
    ! [A3: int,B2: int,P3: int,Q3: int] :
      ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P3),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).

% normalize_eq
tff(fact_6160_the__inv__into__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,B),X5: B] : the_inv_into(A,B,A5,F3,X5) = the(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lf(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A5),F3),X5)) ).

% the_inv_into_def
tff(fact_6161_quotient__of__Fract,axiom,
    ! [A3: int,B2: int] : quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) ).

% quotient_of_Fract
tff(fact_6162_Fract__less__zero__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).

% Fract_less_zero_iff
tff(fact_6163_zero__less__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).

% zero_less_Fract_iff
tff(fact_6164_Fract__less__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B2)) ) ) ).

% Fract_less_one_iff
tff(fact_6165_one__less__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A3)) ) ) ).

% one_less_Fract_iff
tff(fact_6166_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),X: B,B4: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( pp(aa(set(B),bool,member(B,X),aa(set(A),set(B),image2(A,B,F3),A5)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => pp(aa(set(A),bool,member(A,the_inv_into(A,B,A5,F3,X)),B4)) ) ) ) ).

% the_inv_into_into
tff(fact_6167_zero__le__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).

% zero_le_Fract_iff
tff(fact_6168_Fract__le__zero__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).

% Fract_le_zero_iff
tff(fact_6169_one__le__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3)) ) ) ).

% one_le_Fract_iff
tff(fact_6170_Fract__le__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B2)) ) ) ).

% Fract_le_one_iff
tff(fact_6171_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G3: fun(A,B),C4: set(A),B4: set(A),X: A] :
      ( inj_on(A,B,G3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))
       => pp(aa(set(fun(B,A)),bool,member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_agj(fun(A,B),fun(set(A),fun(A,fun(B,A))),G3),C4),X)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6172_positive__rat,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(rat,bool,positive,aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B2))) ) ).

% positive_rat
tff(fact_6173_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A5: set(B)] :
      ( ( B4 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A5,B4) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_6174_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A5,B4) = bot_bot(set(fun(A,B))) )
    <=> ( ( A5 != bot_bot(set(A)) )
        & ( B4 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_6175_Func__empty,axiom,
    ! [B: $tType,A: $tType,B4: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B4) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert(fun(A,B)),aTP_Lamp_agk(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_6176_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
    <=> pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y),X))) ) ).

% less_rat_def
tff(fact_6177_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A16: set(B),B15: set(A),F22: fun(C,D),B23: set(C),A24: set(D)] :
      ( ( aa(set(B),set(A),image2(B,A,F1),A16) = B15 )
     => ( inj_on(C,D,F22,B23)
       => ( pp(aa(set(D),bool,aa(set(D),fun(set(D),bool),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F22),B23)),A24))
         => ( ( ( B23 = bot_bot(set(C)) )
             => ( A24 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B23,B15) = aa(set(fun(D,B)),set(fun(C,A)),image2(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B23,F1,F22)),bNF_Wellorder_Func(D,B,A24,A16)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_6178_positive__def,axiom,
    positive = aa(fun(product_prod(int,int),bool),fun(rat,bool),map_fun(rat,product_prod(int,int),bool,bool,rep_Rat,id(bool)),aTP_Lamp_agl(product_prod(int,int),bool)) ).

% positive_def
tff(fact_6179_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G3: fun(A,B),A24: set(A),A16: set(B),F1: fun(B,C),B15: set(C),F22: fun(D,A),B23: set(D)] :
      ( pp(aa(set(fun(A,B)),bool,member(fun(A,B),G3),bNF_Wellorder_Func(A,B,A24,A16)))
     => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F1),A16)),B15))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,F22),B23)),A24))
         => pp(aa(set(fun(D,C)),bool,member(fun(D,C),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B23,F1,F22),G3)),bNF_Wellorder_Func(D,C,B23,B15))) ) ) ) ).

% Func_map
tff(fact_6180_positive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa(rat,bool,positive,X))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,X))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X))))) ) ).

% positive.rep_eq
tff(fact_6181_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_agm(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_6182_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_agn(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_6183_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_6184_Fract_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(int,rat,aa(int,fun(int,rat),fract,Xa2),X) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),X),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa2),X))) ).

% Fract.abs_eq
tff(fact_6185_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_6186_Fract__def,axiom,
    fract = aa(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,rat)),map_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),id(int),map_fun(int,int,product_prod(int,int),rat,id(int),abs_Rat)),aTP_Lamp_ago(int,fun(int,product_prod(int,int)))) ).

% Fract_def
tff(fact_6187_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_agp(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_6188_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_agq(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_6189_of__rat__def,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A) = aa(fun(product_prod(int,int),A),fun(rat,A),map_fun(rat,product_prod(int,int),A,A,rep_Rat,id(A)),aTP_Lamp_agr(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_6190_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa2),Xa2))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa2)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_6191_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3)) ) ) ).

% zero_less_of_rat_iff
tff(fact_6192_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),zero_zero(rat))) ) ) ).

% of_rat_less_0_iff
tff(fact_6193_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R3)) ) ) ).

% zero_le_of_rat_iff
tff(fact_6194_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),zero_zero(rat))) ) ) ).

% of_rat_le_0_iff
tff(fact_6195_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R3)) ) ) ).

% one_less_of_rat_iff
tff(fact_6196_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),one_one(rat))) ) ) ).

% of_rat_less_1_iff
tff(fact_6197_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R3)) ) ) ).

% one_le_of_rat_iff
tff(fact_6198_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),one_one(rat))) ) ) ).

% of_rat_le_1_iff
tff(fact_6199_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat,S2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),aa(rat,A,field_char_0_of_rat(A),S2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),S2)) ) ) ).

% of_rat_less
tff(fact_6200_of__rat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F3: fun(B,rat),A5: set(B)] : aa(rat,A,field_char_0_of_rat(A),aa(set(B),rat,aa(fun(B,rat),fun(set(B),rat),groups7121269368397514597t_prod(B,rat),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ags(fun(B,rat),fun(B,A),F3)),A5) ) ).

% of_rat_prod
tff(fact_6201_of__rat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F3: fun(B,rat),A5: set(B)] : aa(rat,A,field_char_0_of_rat(A),aa(set(B),rat,aa(fun(B,rat),fun(set(B),rat),groups7311177749621191930dd_sum(B,rat),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ags(fun(B,rat),fun(B,A),F3)),A5) ) ).

% of_rat_sum
tff(fact_6202_one__rat_Orsp,axiom,
    pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),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_6203_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat,S2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),aa(rat,A,field_char_0_of_rat(A),S2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),S2)) ) ) ).

% of_rat_less_eq
tff(fact_6204_of__rat__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_mult
tff(fact_6205_zero__rat_Orsp,axiom,
    pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),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_6206_ratrel__def,axiom,
    ! [X5: product_prod(int,int),Xa3: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X5),Xa3))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X5) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Xa3) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X5)),aa(product_prod(int,int),int,product_snd(int,int),Xa3)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa3)),aa(product_prod(int,int),int,product_snd(int,int),X5)) ) ) ) ).

% ratrel_def
tff(fact_6207_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(product_prod(int,int),int,product_snd(int,int),X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_6208_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa2),Xa2))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa2)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_6209_positive_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( pp(aa(rat,bool,positive,aa(product_prod(int,int),rat,abs_Rat,X)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ).

% positive.abs_eq
tff(fact_6210_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),X)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_6211_cr__rat__def,axiom,
    ! [X5: product_prod(int,int),Xa3: rat] :
      ( cr_rat(X5,Xa3)
    <=> ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X5),X5))
        & ( aa(product_prod(int,int),rat,abs_Rat,X5) = Xa3 ) ) ) ).

% cr_rat_def
tff(fact_6212_Code__Numeral_Odup__def,axiom,
    code_dup = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),aTP_Lamp_agt(int,int)) ).

% Code_Numeral.dup_def
tff(fact_6213_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_agu(rat,fun(product_prod(int,int),bool),X)) ).

% quotient_of_def
tff(fact_6214_ntrancl__Suc,axiom,
    ! [A: $tType,N2: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,N2),R2) = relcomp(A,A,A,transitive_ntrancl(A,N2,R2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),id2(A)),R2)) ).

% ntrancl_Suc
tff(fact_6215_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( algebr8660921524188924756oprime(A,C2,A3)
            & algebr8660921524188924756oprime(A,C2,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_6216_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
        <=> ( algebr8660921524188924756oprime(A,A3,C2)
            & algebr8660921524188924756oprime(A,B2,C2) ) ) ) ).

% coprime_mult_left_iff
tff(fact_6217_pair__in__Id__conv,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
    <=> ( A3 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_6218_IdI,axiom,
    ! [A: $tType,A3: A] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id2(A))) ).

% IdI
tff(fact_6219_Id__O__R,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : relcomp(A,A,B,id2(A),R2) = R2 ).

% Id_O_R
tff(fact_6220_R__O__Id,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : relcomp(A,B,B,R2,id2(B)) = R2 ).

% R_O_Id
tff(fact_6221_bijective__Id,axiom,
    ! [A: $tType] : bijective(A,A,id2(A)) ).

% bijective_Id
tff(fact_6222_rtrancl__empty,axiom,
    ! [A: $tType] : transitive_rtrancl(A,bot_bot(set(product_prod(A,A)))) = id2(A) ).

% rtrancl_empty
tff(fact_6223_rtrancl__reflcl,axiom,
    ! [A: $tType,R2: 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))),R2),id2(A))) = transitive_rtrancl(A,R2) ).

% rtrancl_reflcl
tff(fact_6224_rtrancl__reflcl__absorb,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),id2(A)) = transitive_rtrancl(A,R2) ).

% rtrancl_reflcl_absorb
tff(fact_6225_total__on__diff__Id,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( total_on(A,A5,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))),R3),id2(A)))
    <=> total_on(A,A5,R3) ) ).

% total_on_diff_Id
tff(fact_6226_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_6227_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_6228_trancl__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),id2(A))) = transitive_rtrancl(A,R3) ).

% trancl_reflcl
tff(fact_6229_normalize__stable,axiom,
    ! [Q3: int,P3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q3))
     => ( algebr8660921524188924756oprime(int,P3,Q3)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) ) ) ) ).

% normalize_stable
tff(fact_6230_prod__coprime__left,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(B),F3: fun(B,A),A3: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => algebr8660921524188924756oprime(A,aa(B,A,F3,I3),A3) )
         => algebr8660921524188924756oprime(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5),A3) ) ) ).

% prod_coprime_left
tff(fact_6231_prod__coprime__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(B),A3: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A5))
             => algebr8660921524188924756oprime(A,A3,aa(B,A,F3,I3)) )
         => algebr8660921524188924756oprime(A,A3,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5)) ) ) ).

% prod_coprime_right
tff(fact_6232_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_6233_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_6234_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_6235_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_6236_quotient__of__coprime,axiom,
    ! [R3: rat,P3: int,Q3: int] :
      ( ( quotient_of(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => algebr8660921524188924756oprime(int,P3,Q3) ) ).

% quotient_of_coprime
tff(fact_6237_IdE,axiom,
    ! [A: $tType,P3: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),id2(A)))
     => ~ ! [X3: A] : P3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).

% IdE
tff(fact_6238_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
     => ( A3 = B2 ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_6239_normalize__coprime,axiom,
    ! [R3: product_prod(int,int),P3: int,Q3: int] :
      ( ( normalize(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => algebr8660921524188924756oprime(int,P3,Q3) ) ).

% normalize_coprime
tff(fact_6240_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_6241_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_6242_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2))
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2)) ) ) ) ) ).

% divides_mult
tff(fact_6243_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A3: A,N2: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),N2),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),N2),M) )
         => ( algebr8660921524188924756oprime(A,M,N2)
           => ( modulo_modulo(A,A3,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_6244_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [N2: A,A3: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N2),A3),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N2),B2),M) )
         => ( algebr8660921524188924756oprime(A,M,N2)
           => ( modulo_modulo(A,A3,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_6245_Id__def,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_agv(product_prod(A,A),bool)) ).

% Id_def
tff(fact_6246_Id__fstsnd__eq,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_agw(product_prod(A,A),bool)) ).

% Id_fstsnd_eq
tff(fact_6247_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,C2) ) ) ).

% invertible_coprime
tff(fact_6248_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,A4: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
           => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
             => algebr8660921524188924756oprime(A,A4,B3) ) ) ) ) ).

% gcd_coprime
tff(fact_6249_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
         => ? [A10: A,B9: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A10),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B9),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & algebr8660921524188924756oprime(A,A10,B9) ) ) ) ).

% gcd_coprime_exists
tff(fact_6250_Rat__induct,axiom,
    ! [P: fun(rat,bool),Q3: rat] :
      ( ! [A6: int,B5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
         => ( algebr8660921524188924756oprime(int,A6,B5)
           => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A6),B5))) ) )
     => pp(aa(rat,bool,P,Q3)) ) ).

% Rat_induct
tff(fact_6251_Rat__cases,axiom,
    ! [Q3: rat] :
      ~ ! [A6: int,B5: int] :
          ( ( Q3 = aa(int,rat,aa(int,fun(int,rat),fract,A6),B5) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
           => ~ algebr8660921524188924756oprime(int,A6,B5) ) ) ).

% Rat_cases
tff(fact_6252_rtrancl__trancl__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_rtrancl(A,R3) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R3)),id2(A)) ).

% rtrancl_trancl_reflcl
tff(fact_6253_rtrancl__unfold,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_rtrancl(A,R3) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),id2(A)),relcomp(A,A,A,transitive_rtrancl(A,R3),R3)) ).

% rtrancl_unfold
tff(fact_6254_reflcl__set__eq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X5: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)),fequal(A)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),id2(A)))) ) ).

% reflcl_set_eq
tff(fact_6255_Rat__cases__nonzero,axiom,
    ! [Q3: rat] :
      ( ! [A6: int,B5: int] :
          ( ( Q3 = aa(int,rat,aa(int,fun(int,rat),fract,A6),B5) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
           => ( ( A6 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A6,B5) ) ) )
     => ( Q3 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_6256_rtrancl__Int__subset,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id2(A)),S2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_rtrancl(A,R3)),S2),R3)),S2))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),S2)) ) ) ).

% rtrancl_Int_subset
tff(fact_6257_quotient__of__unique,axiom,
    ! [R3: rat] :
    ? [X3: product_prod(int,int)] :
      ( ( R3 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X3)),aa(product_prod(int,int),int,product_snd(int,int),X3)) )
      & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),X3)))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X3),aa(product_prod(int,int),int,product_snd(int,int),X3))
      & ! [Y4: product_prod(int,int)] :
          ( ( ( R3 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y4)),aa(product_prod(int,int),int,product_snd(int,int),Y4)) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y4)))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y4),aa(product_prod(int,int),int,product_snd(int,int),Y4)) )
         => ( Y4 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_6258_relInvImage__Id__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(B)] :
      ( ! [A12: A,A23: A] :
          ( ( aa(A,B,F3,A12) = aa(A,B,F3,A23) )
        <=> ( A12 = A23 ) )
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),bNF_Gr7122648621184425601vImage(A,B,A5,id_on(B,B4),F3)),id2(A))) ) ).

% relInvImage_Id_on
tff(fact_6259_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),field_char_0_Rats(A)))
         => ~ ! [A6: int,B5: int] :
                ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
               => ( algebr8660921524188924756oprime(int,A6,B5)
                 => ( X != aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A6)),aa(int,A,ring_1_of_int(A),B5)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_6260_Rats__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),field_char_0_Rats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),field_char_0_Rats(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_mult
tff(fact_6261_relInvImage__mono,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),A5: set(B),F3: fun(B,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
     => pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr7122648621184425601vImage(B,A,A5,R1,F3)),bNF_Gr7122648621184425601vImage(B,A,A5,R22,F3))) ) ).

% relInvImage_mono
tff(fact_6262_Ints__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ring_1_Ints(A)),field_char_0_Rats(A))) ) ).

% Ints_subset_Rats
tff(fact_6263_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)),N2) ) ).

% coprime_diff_one_left_nat
tff(fact_6264_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
     => algebr8660921524188924756oprime(nat,N2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_6265_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),A5: set(A),G3: fun(B,nat),B4: set(B)] :
      ( inj_on(A,nat,F3,A5)
     => ( inj_on(B,nat,G3,B4)
       => ( ! [A6: A,B5: B] :
              ( pp(aa(set(A),bool,member(A,A6),A5))
             => ( pp(aa(set(B),bool,member(B,B5),B4))
               => algebr8660921524188924756oprime(nat,aa(A,nat,F3,A6),aa(B,nat,G3,B5)) ) )
         => inj_on(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_agx(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F3),G3)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_6266_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R2: set(product_prod(B,B)),F3: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A5,R2,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_agy(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),A5),R2),F3)) ).

% relInvImage_def
tff(fact_6267_relImage__relInvImage,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F3),A5),aa(set(B),fun(A,set(A)),aTP_Lamp_xx(fun(B,A),fun(set(B),fun(A,set(A))),F3),A5))))
     => ( bNF_Gr4221423524335903396lImage(B,A,bNF_Gr7122648621184425601vImage(B,A,A5,R2,F3),F3) = R2 ) ) ).

% relImage_relInvImage
tff(fact_6268_relInvImage__UNIV__relImage,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F3: fun(A,B)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),bNF_Gr7122648621184425601vImage(A,B,top_top(set(A)),bNF_Gr4221423524335903396lImage(A,B,R2,F3),F3))) ).

% relInvImage_UNIV_relImage
tff(fact_6269_relImage__mono,axiom,
    ! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
     => pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr4221423524335903396lImage(A,B,R1,F3)),bNF_Gr4221423524335903396lImage(A,B,R22,F3))) ) ).

% relImage_mono
tff(fact_6270_relImage__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F3: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R2,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_agz(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),R2),F3)) ).

% relImage_def
tff(fact_6271_card__quotient__disjoint,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( inj_on(A,set(set(A)),aTP_Lamp_aha(set(product_prod(A,A)),fun(A,set(set(A))),R3),A5)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A5,R3)) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_quotient_disjoint
tff(fact_6272_eq__f__restr__ss__eq,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(fun(A,option(B)),fun(A,option(B))),A5: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A5))))
     => ( ( A5 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A5),aa(set(A),set(A),uminus_uminus(set(A)),S2)) )
      <=> ( map_le(A,B,A5,aa(fun(A,option(B)),fun(A,option(B)),F3,A5))
          & ( 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)),F3,A5))),dom(A,B,A5)) ) ) ) ) ).

% eq_f_restr_ss_eq
tff(fact_6273_quotient__empty,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : equiv_quotient(A,bot_bot(set(A)),R3) = bot_bot(set(set(A))) ).

% quotient_empty
tff(fact_6274_quotient__is__empty,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( ( equiv_quotient(A,A5,R3) = bot_bot(set(set(A))) )
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% quotient_is_empty
tff(fact_6275_quotient__is__empty2,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( ( bot_bot(set(set(A))) = equiv_quotient(A,A5,R3) )
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% quotient_is_empty2
tff(fact_6276_map__leI,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ! [X3: A,V3: B] :
          ( ( aa(A,option(B),M1,X3) = aa(B,option(B),some(B),V3) )
         => ( aa(A,option(B),M22,X3) = aa(B,option(B),some(B),V3) ) )
     => map_le(A,B,M1,M22) ) ).

% map_leI
tff(fact_6277_map__leD,axiom,
    ! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),K: A,V2: B] :
      ( map_le(A,B,M1,M22)
     => ( ( aa(A,option(B),M1,K) = aa(B,option(B),some(B),V2) )
       => ( aa(A,option(B),M22,K) = aa(B,option(B),some(B),V2) ) ) ) ).

% map_leD
tff(fact_6278_map__le__empty,axiom,
    ! [B: $tType,A: $tType,G3: fun(A,option(B))] : map_le(A,B,aTP_Lamp_bk(A,option(B)),G3) ).

% map_le_empty
tff(fact_6279_map__le__implies__dom__le,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( map_le(A,B,F3,G3)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),dom(A,B,F3)),dom(A,B,G3))) ) ).

% map_le_implies_dom_le
tff(fact_6280_quotient__diff1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),A3: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_aha(set(product_prod(A,A)),fun(A,set(set(A))),R3),A5)
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => ( equiv_quotient(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))),R3) = 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,A5,R3)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))),R3)) ) ) ) ).

% quotient_diff1
tff(fact_6281_finite__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
       => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
         => pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).

% finite_equiv_class
tff(fact_6282_map__le__imp__upd__le,axiom,
    ! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X: A,Y: B] :
      ( map_le(A,B,M1,M22)
     => map_le(A,B,fun_upd(A,option(B),M1,X,none(B)),fun_upd(A,option(B),M22,X,aa(B,option(B),some(B),Y))) ) ).

% map_le_imp_upd_le
tff(fact_6283_finite__quotient,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
       => pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A5,R3))) ) ) ).

% finite_quotient
tff(fact_6284_eq__f__restr__conv,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(fun(A,option(B)),fun(A,option(B))),A5: fun(A,option(B))] :
      ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A5))))
        & ( A5 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A5),aa(set(A),set(A),uminus_uminus(set(A)),S2)) ) )
    <=> ( map_le(A,B,A5,aa(fun(A,option(B)),fun(A,option(B)),F3,A5))
        & ( 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)),F3,A5))),dom(A,B,A5)) ) ) ) ).

% eq_f_restr_conv
tff(fact_6285_le__map__mmupd__not__dom,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K5: set(A),V2: B] : map_le(A,B,M,map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K5),dom(A,B,M)),V2)) ).

% le_map_mmupd_not_dom
tff(fact_6286_quotient__def,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] : equiv_quotient(A,A5,R3) = 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_ahb(set(product_prod(A,A)),fun(A,set(set(A))),R3)),A5)) ).

% quotient_def
tff(fact_6287_mmupd__notin__upd,axiom,
    ! [B: $tType,A: $tType,K: A,K5: set(A),M: fun(A,option(B)),V2: B] :
      ( ~ pp(aa(set(A),bool,member(A,K),K5))
     => ( aa(A,option(B),map_mmupd(A,B,M,K5,V2),K) = aa(A,option(B),M,K) ) ) ).

% mmupd_notin_upd
tff(fact_6288_ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R3: set(product_prod(A,B)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image(A,B,R3),A5))) ) ) ).

% ImageI
tff(fact_6289_Image__empty2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,A))] : aa(set(B),set(A),image(B,A,R2),bot_bot(set(B))) = bot_bot(set(A)) ).

% Image_empty2
tff(fact_6290_Image__Id,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),image(A,A,id2(A)),A5) = A5 ).

% Image_Id
tff(fact_6291_map__mmupd__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),V2: B] : map_mmupd(A,B,M,bot_bot(set(A)),V2) = M ).

% map_mmupd_empty
tff(fact_6292_mmupd__in__upd,axiom,
    ! [A: $tType,B: $tType,K: A,K5: set(A),M: fun(A,option(B)),V2: B] :
      ( pp(aa(set(A),bool,member(A,K),K5))
     => ( aa(A,option(B),map_mmupd(A,B,M,K5,V2),K) = aa(B,option(B),some(B),V2) ) ) ).

% mmupd_in_upd
tff(fact_6293_Image__empty1,axiom,
    ! [B: $tType,A: $tType,X6: set(B)] : aa(set(B),set(A),image(B,A,bot_bot(set(product_prod(B,A)))),X6) = bot_bot(set(A)) ).

% Image_empty1
tff(fact_6294_Image__Id__on,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : aa(set(A),set(A),image(A,A,id_on(A,A5)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) ).

% Image_Id_on
tff(fact_6295_dom__mmupd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K5: set(A),V2: B] : dom(A,B,map_mmupd(A,B,M,K5,V2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,M)),K5) ).

% dom_mmupd
tff(fact_6296_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R3: set(product_prod(B,A)),A3: B] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R3)) ) ).

% Image_singleton_iff
tff(fact_6297_pair__vimage__is__Image,axiom,
    ! [A: $tType,B: $tType,U: B,E5: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U)),E5) = aa(set(B),set(A),image(B,A,E5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),U),bot_bot(set(B)))) ).

% pair_vimage_is_Image
tff(fact_6298_Image__Int__subset,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A5: set(B),B4: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,R2),A5)),aa(set(B),set(A),image(B,A,R2),B4)))) ).

% Image_Int_subset
tff(fact_6299_Image__mono,axiom,
    ! [B: $tType,A: $tType,R6: set(product_prod(A,B)),R3: set(product_prod(A,B)),A15: set(A),A5: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R6),R3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A15),A5))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,R6),A15)),aa(set(A),set(B),image(A,B,R3),A5))) ) ) ).

% Image_mono
tff(fact_6300_Image__closed__trancl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),X6)),X6))
     => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R3)),X6) = X6 ) ) ).

% Image_closed_trancl
tff(fact_6301_rtrancl__image__unfold__right,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),V: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E5),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),V))),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),V))) ).

% rtrancl_image_unfold_right
tff(fact_6302_rtrancl__reachable__induct,axiom,
    ! [A: $tType,I5: set(A),INV: set(A),E5: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),I5),INV))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E5),INV)),INV))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),I5)),INV)) ) ) ).

% rtrancl_reachable_induct
tff(fact_6303_Un__Image,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S: set(product_prod(B,A)),A5: set(B)] : aa(set(B),set(A),image(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),R2),S)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,R2),A5)),aa(set(B),set(A),image(B,A,S),A5)) ).

% Un_Image
tff(fact_6304_Image__Un,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A5: set(B),B4: set(B)] : aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,R2),A5)),aa(set(B),set(A),image(B,A,R2),B4)) ).

% Image_Un
tff(fact_6305_rtrancl__image__advance__rtrancl,axiom,
    ! [A: $tType,Q3: A,R2: set(product_prod(A,A)),Q0: set(A),X: A] :
      ( pp(aa(set(A),bool,member(A,Q3),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X)),transitive_rtrancl(A,R2)))
       => pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0))) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_6306_rtrancl__image__advance,axiom,
    ! [A: $tType,Q3: A,R2: set(product_prod(A,A)),Q0: set(A),X: A] :
      ( pp(aa(set(A),bool,member(A,Q3),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X)),R2))
       => pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0))) ) ) ).

% rtrancl_image_advance
tff(fact_6307_ImageE,axiom,
    ! [A: $tType,B: $tType,B2: A,R3: set(product_prod(B,A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R3),A5)))
     => ~ ! [X3: B] :
            ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),B2)),R3))
           => ~ pp(aa(set(B),bool,member(B,X3),A5)) ) ) ).

% ImageE
tff(fact_6308_Image__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R3: set(product_prod(B,A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R3),A5)))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
          & pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),B2)),R3)) ) ) ).

% Image_iff
tff(fact_6309_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),B2: B,R3: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
       => pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image(A,B,R3),A5))) ) ) ).

% rev_ImageI
tff(fact_6310_relcomp__Image,axiom,
    ! [A: $tType,C: $tType,B: $tType,X6: set(product_prod(B,C)),Y6: set(product_prod(C,A)),Z4: set(B)] : aa(set(B),set(A),image(B,A,relcomp(B,C,A,X6,Y6)),Z4) = aa(set(C),set(A),image(C,A,Y6),aa(set(B),set(C),image(B,C,X6),Z4)) ).

% relcomp_Image
tff(fact_6311_trancl__Image__unfold__left,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),S: set(A)] : aa(set(A),set(A),image(A,A,transitive_trancl(A,E5)),S) = aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),aa(set(A),set(A),image(A,A,E5),S)) ).

% trancl_Image_unfold_left
tff(fact_6312_trancl__Image__unfold__right,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),S: set(A)] : aa(set(A),set(A),image(A,A,transitive_trancl(A,E5)),S) = aa(set(A),set(A),image(A,A,E5),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),S)) ).

% trancl_Image_unfold_right
tff(fact_6313_map__mmupdE,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),K5: set(B),V2: A,K: B,X: A] :
      ( ( aa(B,option(A),map_mmupd(B,A,M,K5,V2),K) = aa(A,option(A),some(A),X) )
     => ( ( ~ pp(aa(set(B),bool,member(B,K),K5))
         => ( aa(B,option(A),M,K) != aa(A,option(A),some(A),X) ) )
       => ~ ( pp(aa(set(B),bool,member(B,K),K5))
           => ( X != V2 ) ) ) ) ).

% map_mmupdE
tff(fact_6314_map__mmupd__def,axiom,
    ! [A: $tType,B: $tType,K: B,K5: set(B),M: fun(B,option(A)),V2: A] :
      ( ( pp(aa(set(B),bool,member(B,K),K5))
       => ( aa(B,option(A),map_mmupd(B,A,M,K5,V2),K) = aa(A,option(A),some(A),V2) ) )
      & ( ~ pp(aa(set(B),bool,member(B,K),K5))
       => ( aa(B,option(A),map_mmupd(B,A,M,K5,V2),K) = aa(B,option(A),M,K) ) ) ) ).

% map_mmupd_def
tff(fact_6315_finite__Image,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R2))
     => pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),image(A,B,R2),A5))) ) ).

% finite_Image
tff(fact_6316_finite__Image__subset,axiom,
    ! [A: $tType,B: $tType,A5: set(product_prod(B,A)),B4: set(B),C4: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image(B,A,A5),B4)))
     => ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),C4),A5))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image(B,A,C4),B4))) ) ) ).

% finite_Image_subset
tff(fact_6317_Image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(B,A)),B4: fun(C,set(B)),A5: set(C)] : aa(set(B),set(A),image(B,A,R3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ahc(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R3),B4)),A5)) ).

% Image_UN
tff(fact_6318_Image__empty__rtrancl__Image__id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),V2: A] :
      ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A))) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_6319_Image__empty__trancl__Image__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),V2: A] :
      ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),image(A,A,transitive_trancl(A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_6320_reachable__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R8: set(product_prod(A,A)),X6: set(A),X11: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),R8))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),X11))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),X6)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R8)),X11))) ) ) ).

% reachable_mono
tff(fact_6321_quotientI,axiom,
    ! [A: $tType,X: A,A5: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => pp(aa(set(set(A)),bool,member(set(A),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),equiv_quotient(A,A5,R3))) ) ).

% quotientI
tff(fact_6322_quotientE,axiom,
    ! [A: $tType,X6: set(A),A5: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
     => ~ ! [X3: A] :
            ( ( X6 = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A)))) )
           => ~ pp(aa(set(A),bool,member(A,X3),A5)) ) ) ).

% quotientE
tff(fact_6323_Image__subset__snd__image,axiom,
    ! [A: $tType,B: $tType,A5: set(product_prod(B,A)),B4: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,A5),B4)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A5))) ).

% Image_subset_snd_image
tff(fact_6324_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)),aa(set(A),set(A),image(A,A,transitive_trancl(A,E5)),Vi)),Vi) = aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),Vi) ).

% trancl_image_by_rtrancl
tff(fact_6325_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),A3: B] : aa(set(B),set(A),image(B,A,R3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B)))) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_bq(set(product_prod(B,A)),fun(B,fun(A,bool)),R3),A3)) ).

% Image_singleton
tff(fact_6326_Image__subset,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B)),A5: set(A),B4: set(B),C4: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,R3),C4)),B4)) ) ).

% Image_subset
tff(fact_6327_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(B,A)),B4: fun(C,set(B)),A5: set(C)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A5)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ahc(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R3),B4)),A5)))) ).

% Image_INT_subset
tff(fact_6328_rtrancl__apply__insert,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,S: set(A)] : aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ).

% rtrancl_apply_insert
tff(fact_6329_E__closed__restr__reach__cases,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,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_rtrancl(A,E5)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E5),R2)),R2))
       => ( ~ pp(aa(set(A),bool,member(A,V2),R2))
         => ~ ( ~ pp(aa(set(A),bool,member(A,U),R2))
             => ~ pp(aa(set(product_prod(A,A)),bool,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_rtrancl(A,rel_restrict(A,E5,R2)))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_6330_rel__restrict__tranclI,axiom,
    ! [A: $tType,X: A,Y: A,E5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,E5)))
     => ( ~ pp(aa(set(A),bool,member(A,X),R2))
       => ( ~ pp(aa(set(A),bool,member(A,Y),R2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E5),R2)),R2))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,rel_restrict(A,E5,R2)))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_6331_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),B4: set(B)] : aa(set(B),set(A),image(B,A,R3),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ahd(set(product_prod(B,A)),fun(B,set(A)),R3)),B4)) ).

% Image_eq_UN
tff(fact_6332_Sigma__Image,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B4: fun(B,set(A)),X6: set(B)] : aa(set(B),set(A),image(B,A,product_Sigma(B,A,A5,B4)),X6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X6),A5))) ).

% Sigma_Image
tff(fact_6333_UN__Image,axiom,
    ! [B: $tType,A: $tType,C: $tType,X6: fun(C,set(product_prod(B,A))),I5: set(C),S: set(B)] : aa(set(B),set(A),image(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),X6),I5))),S) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_ahe(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),X6),S)),I5)) ).

% UN_Image
tff(fact_6334_finite__reachable__advance,axiom,
    ! [A: $tType,E5: set(product_prod(A,A)),V0: A,V2: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V0),bot_bot(set(A))))))
     => ( pp(aa(set(product_prod(A,A)),bool,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)),transitive_rtrancl(A,E5)))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))))) ) ) ).

% finite_reachable_advance
tff(fact_6335_rtrancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))))) ) ).

% rtrancl_Image_advance_ss
tff(fact_6336_trancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_trancl(A,E5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,transitive_trancl(A,E5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))))) ) ).

% trancl_Image_advance_ss
tff(fact_6337_trancl__restrict__reachable,axiom,
    ! [A: $tType,U: A,V2: A,E5: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,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)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E5),S)),S))
       => ( pp(aa(set(A),bool,member(A,U),S))
         => pp(aa(set(product_prod(A,A)),bool,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_yj(set(A),fun(A,set(A)),S)))))) ) ) ) ).

% trancl_restrict_reachable
tff(fact_6338_singleton__quotient,axiom,
    ! [A: $tType,X: A,R3: set(product_prod(A,A))] : equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),R3) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% singleton_quotient
tff(fact_6339_Image__fold,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R2))
     => ( aa(set(A),set(B),image(A,B,R2),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_xl(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R2) ) ) ).

% Image_fold
tff(fact_6340_map__mmupd__update__less,axiom,
    ! [A: $tType,B: $tType,K5: set(A),K7: set(A),M: fun(A,option(B)),V2: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K5),K7))
     => map_le(A,B,map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K5),dom(A,B,M)),V2),map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K7),dom(A,B,M)),V2)) ) ).

% map_mmupd_update_less
tff(fact_6341_listrel__Cons,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),X: B,Xs: list(B)] : aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R3)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),bot_bot(set(list(B))))) = set_Cons(A,aa(set(B),set(A),image(B,A,R3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))),aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R3)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert(list(B)),Xs),bot_bot(set(list(B)))))) ).

% listrel_Cons
tff(fact_6342_positive_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,ratrel,fequal(bool)),aTP_Lamp_agl(product_prod(int,int),bool)),aTP_Lamp_agl(product_prod(int,int),bool))) ).

% positive.rsp
tff(fact_6343_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),listrel(A,A,transitive_rtrancl(A,R3)))) ).

% listrel_rtrancl_refl
tff(fact_6344_listrel__Nil,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(B,A))] : aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R3)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert(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))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listrel_Nil
tff(fact_6345_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),times_times(A)),times_times(B)))
           => pp(aa(fun(B,fun(nat,B)),bool,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),bool),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R2,bNF_rel_fun(nat,nat,A,B,fequal(nat),R2)),power_power(A)),power_power(B))) ) ) ) ).

% power_transfer
tff(fact_6346_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),nil(B))),listrel(A,B,R3)))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_6347_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xs)),listrel(A,B,R3)))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_6348_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))),listrel(A,B,R3))) ).

% listrel.Nil
tff(fact_6349_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys2)),listrel(A,B,R3)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) ) ) ).

% listrel_eq_len
tff(fact_6350_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A)),Zs: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel(A,A,transitive_rtrancl(A,R3))))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys2),Zs)),listrel(A,A,transitive_rtrancl(A,R3))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),listrel(A,A,transitive_rtrancl(A,R3)))) ) ) ).

% listrel_rtrancl_trans
tff(fact_6351_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S2))
     => pp(aa(set(product_prod(list(A),list(B))),bool,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),bool),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R3)),listrel(A,B,S2))) ) ).

% listrel_mono
tff(fact_6352_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys2: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2))),listrel(A,B,R3)))
     => ~ ! [X3: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
           => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y)),R3))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys2)),listrel(A,B,R3))) ) ) ) ).

% listrel_Cons2
tff(fact_6353_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys2: list(A),Xs: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)),Xs)),listrel(A,B,R3)))
     => ~ ! [Y3: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
           => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y3)),R3))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys2),Ys3)),listrel(A,B,R3))) ) ) ) ).

% listrel_Cons1
tff(fact_6354_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,R3: set(product_prod(A,B)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R3))
     => ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys2)),listrel(A,B,R3)))
       => pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2))),listrel(A,B,R3))) ) ) ).

% listrel.Cons
tff(fact_6355_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel1(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel(A,A,transitive_rtrancl(A,R3)))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_6356_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),listrel(A,A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),transitive_rtrancl(list(A),listrel1(A,R3)))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_6357_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R3)))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X3: A,Y3: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
                 => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R3))
                   => ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys3)),listrel(A,B,R3))) ) ) ) ) ) ).

% listrel.cases
tff(fact_6358_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R3)))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X4: A,Y5: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
            & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y5)),R3))
            & pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys4)),listrel(A,B,R3))) ) ) ) ).

% listrel.simps
tff(fact_6359_rel__fun__Collect__case__prodD,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,A5: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool)),F3: fun(A,C),G3: fun(B,D),X6: set(product_prod(A,B)),X: product_prod(A,B)] :
      ( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A5,B4),F3),G3))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),X6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A5))))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),X6))
         => pp(aa(D,bool,aa(C,fun(D,bool),B4,aa(product_prod(A,B),C,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)),X)),aa(product_prod(A,B),D,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),D),comp(B,D,product_prod(A,B),G3),product_snd(A,B)),X))) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_6360_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),transitive_rtrancl(list(A),listrel1(A,R3)))) ).

% listrel_subset_rtrancl_listrel1
tff(fact_6361_of__rat_Orsp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(fun(product_prod(int,int),A),bool,aa(fun(product_prod(int,int),A),fun(fun(product_prod(int,int),A),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),A,A,ratrel,fequal(A)),aTP_Lamp_agr(product_prod(int,int),A)),aTP_Lamp_agr(product_prod(int,int),A))) ) ).

% of_rat.rsp
tff(fact_6362_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys2)),listrel(A,B,R3)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [N: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N)),aa(nat,B,nth(B,Ys2),N))),R3)) ) ) ) ).

% listrel_iff_nth
tff(fact_6363_fun_Oin__rel,axiom,
    ! [A: $tType,B: $tType,D: $tType,R2: fun(A,fun(B,bool)),A3: fun(D,A),B2: fun(D,B)] :
      ( pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),R2),A3),B2))
    <=> ? [Z6: fun(D,product_prod(A,B))] :
          ( pp(aa(set(fun(D,product_prod(A,B))),bool,member(fun(D,product_prod(A,B)),Z6),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_ahf(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R2))))
          & ( aa(fun(D,product_prod(A,B)),fun(D,A),comp(product_prod(A,B),A,D,product_fst(A,B)),Z6) = A3 )
          & ( aa(fun(D,product_prod(A,B)),fun(D,B),comp(product_prod(A,B),B,D,product_snd(A,B)),Z6) = B2 ) ) ) ).

% fun.in_rel
tff(fact_6364_fun_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: fun(D,A),Y: fun(D,B)] :
      ( pp(aa(fun(D,B),bool,aa(fun(D,C),fun(fun(D,B),bool),bNF_rel_fun(D,D,C,B,fequal(D),Sb),aa(fun(D,A),fun(D,C),comp(A,C,D,I2),X)),Y))
    <=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% fun.rel_map(1)
tff(fact_6365_sub_Orsp,axiom,
    pp(aa(fun(num,fun(num,int)),bool,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),bool),bNF_rel_fun(num,num,fun(num,int),fun(num,int),fequal(num),bNF_rel_fun(num,num,int,int,fequal(num),fequal(int))),aTP_Lamp_ahh(num,fun(num,int))),aTP_Lamp_ahh(num,fun(num,int)))) ).

% sub.rsp
tff(fact_6366_Fract_Orsp,axiom,
    pp(aa(fun(int,fun(int,product_prod(int,int))),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),bool),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_ago(int,fun(int,product_prod(int,int)))),aTP_Lamp_ago(int,fun(int,product_prod(int,int))))) ).

% Fract.rsp
tff(fact_6367_dup_Orsp,axiom,
    pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_agt(int,int)),aTP_Lamp_agt(int,int))) ).

% dup.rsp
tff(fact_6368_less__eq__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less_eq(int)),ord_less_eq(int))) ).

% less_eq_integer.rsp
tff(fact_6369_less__eq__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less_eq(nat)),ord_less_eq(nat))) ).

% less_eq_natural.rsp
tff(fact_6370_less__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less(nat)),ord_less(nat))) ).

% less_natural.rsp
tff(fact_6371_less__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less(int)),ord_less(int))) ).

% less_integer.rsp
tff(fact_6372_times__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_agq(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_agq(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% times_rat.rsp
tff(fact_6373_plus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_agm(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_agm(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% plus_rat.rsp
tff(fact_6374_fun_Omap__ident,axiom,
    ! [A: $tType,D: $tType,T6: fun(D,A)] : aa(fun(D,A),fun(D,A),comp(A,A,D,aTP_Lamp_cq(A,A)),T6) = T6 ).

% fun.map_ident
tff(fact_6375_predicate2__transferD,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),P: fun(A,fun(C,bool)),Q: fun(B,fun(D,bool)),A3: product_prod(A,B),A5: set(product_prod(A,B)),B2: product_prod(C,D),B4: set(product_prod(C,D))] :
      ( pp(aa(fun(B,fun(D,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),bool),bNF_rel_fun(A,B,fun(C,bool),fun(D,bool),R1,bNF_rel_fun(C,D,bool,bool,R22,fequal(bool))),P),Q))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),A3),A5))
       => ( pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),B2),B4))
         => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R1))))
           => ( pp(aa(set(product_prod(C,D)),bool,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),bool),ord_less_eq(set(product_prod(C,D))),B4),aa(fun(product_prod(C,D),bool),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),R22))))
             => ( pp(aa(C,bool,aa(A,fun(C,bool),P,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(C,D),C,product_fst(C,D),B2)))
              <=> pp(aa(D,bool,aa(B,fun(D,bool),Q,aa(product_prod(A,B),B,product_snd(A,B),A3)),aa(product_prod(C,D),D,product_snd(C,D),B2))) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_6376_prod_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T6: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_cq(A,A),aTP_Lamp_aay(B,B)),T6) = T6 ).

% prod.map_ident
tff(fact_6377_uminus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_agp(product_prod(int,int),product_prod(int,int))),aTP_Lamp_agp(product_prod(int,int),product_prod(int,int)))) ).

% uminus_rat.rsp
tff(fact_6378_inverse__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_agn(product_prod(int,int),product_prod(int,int))),aTP_Lamp_agn(product_prod(int,int),product_prod(int,int)))) ).

% inverse_rat.rsp
tff(fact_6379_fun_Orel__mono,axiom,
    ! [D: $tType,B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra2))
     => pp(aa(fun(fun(D,A),fun(fun(D,B),bool)),bool,aa(fun(fun(D,A),fun(fun(D,B),bool)),fun(fun(fun(D,A),fun(fun(D,B),bool)),bool),ord_less_eq(fun(fun(D,A),fun(fun(D,B),bool))),bNF_rel_fun(D,D,A,B,fequal(D),R2)),bNF_rel_fun(D,D,A,B,fequal(D),Ra2))) ) ).

% fun.rel_mono
tff(fact_6380_fun_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: fun(A,fun(C,bool)),X: fun(D,A),G3: fun(B,C),Y: fun(D,B)] :
      ( pp(aa(fun(D,C),bool,aa(fun(D,A),fun(fun(D,C),bool),bNF_rel_fun(D,D,A,C,fequal(D),Sa),X),aa(fun(D,B),fun(D,C),comp(B,C,D,G3),Y)))
    <=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahi(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G3)),X),Y)) ) ).

% fun.rel_map(2)
tff(fact_6381_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => pp(aa(fun(bool,B),bool,aa(fun(bool,A),fun(fun(bool,B),bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),R2),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).

% transfer_rule_of_bool
tff(fact_6382_plus__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_agm(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).

% plus_rat.transfer
tff(fact_6383_one__rat_Otransfer,axiom,
    pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),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_6384_zero__rat_Otransfer,axiom,
    pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),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_6385_Fract_Otransfer,axiom,
    pp(aa(fun(int,fun(int,rat)),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),bool),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_ago(int,fun(int,product_prod(int,int)))),fract)) ).

% Fract.transfer
tff(fact_6386_of__rat_Otransfer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(fun(rat,A),bool,aa(fun(product_prod(int,int),A),fun(fun(rat,A),bool),bNF_rel_fun(product_prod(int,int),rat,A,A,pcr_rat,fequal(A)),aTP_Lamp_agr(product_prod(int,int),A)),field_char_0_of_rat(A))) ) ).

% of_rat.transfer
tff(fact_6387_uminus__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_agp(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat))) ).

% uminus_rat.transfer
tff(fact_6388_times__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_agq(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).

% times_rat.transfer
tff(fact_6389_positive_Otransfer,axiom,
    pp(aa(fun(rat,bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(rat,bool),bool),bNF_rel_fun(product_prod(int,int),rat,bool,bool,pcr_rat,fequal(bool)),aTP_Lamp_agl(product_prod(int,int),bool)),positive)) ).

% positive.transfer
tff(fact_6390_inverse__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_agn(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).

% inverse_rat.transfer
tff(fact_6391_fun__mono,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,C4: fun(A,fun(B,bool)),A5: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool)),D4: fun(C,fun(D,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),C4),A5))
     => ( pp(aa(fun(C,fun(D,bool)),bool,aa(fun(C,fun(D,bool)),fun(fun(C,fun(D,bool)),bool),ord_less_eq(fun(C,fun(D,bool))),B4),D4))
       => pp(aa(fun(fun(A,C),fun(fun(B,D),bool)),bool,aa(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(fun(A,C),fun(fun(B,D),bool)),bool),ord_less_eq(fun(fun(A,C),fun(fun(B,D),bool))),bNF_rel_fun(A,B,C,D,A5,B4)),bNF_rel_fun(A,B,C,D,C4,D4))) ) ) ).

% fun_mono
tff(fact_6392_times__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ra(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).

% times_int.transfer
tff(fact_6393_zero__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),zero_zero(int))) ).

% zero_int.transfer
tff(fact_6394_int__transfer,axiom,
    pp(aa(fun(nat,int),bool,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_ahj(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).

% int_transfer
tff(fact_6395_uminus__int_Otransfer,axiom,
    pp(aa(fun(int,int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),bool),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_rb(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).

% uminus_int.transfer
tff(fact_6396_nat_Otransfer,axiom,
    pp(aa(fun(int,nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),bool),bNF_rel_fun(product_prod(nat,nat),int,nat,nat,pcr_int,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),nat2)) ).

% nat.transfer
tff(fact_6397_one__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),one_one(int))) ).

% one_int.transfer
tff(fact_6398_of__int_Otransfer,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(fun(int,A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(int,A),bool),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_rc(nat,fun(nat,A)))),ring_1_of_int(A))) ) ).

% of_int.transfer
tff(fact_6399_less__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_re(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).

% less_int.transfer
tff(fact_6400_less__eq__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rg(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).

% less_eq_int.transfer
tff(fact_6401_plus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ri(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).

% plus_int.transfer
tff(fact_6402_minus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).

% minus_int.transfer
tff(fact_6403_rel__pred__comp__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),P: fun(B,bool),X5: A] :
      ( rel_pred_comp(A,B,R2,P,X5)
    <=> ? [Y5: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X5),Y5))
          & pp(aa(B,bool,P,Y5)) ) ) ).

% rel_pred_comp_def
tff(fact_6404_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys2)),listrel(A,B,R3)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),X4)) ) ) ) ).

% listrel_iff_zip
tff(fact_6405_ball__empty,axiom,
    ! [A: $tType,P: fun(A,bool),X5: A] :
      ( pp(aa(set(A),bool,member(A,X5),bot_bot(set(A))))
     => pp(aa(A,bool,P,X5)) ) ).

% ball_empty
tff(fact_6406_INF__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_ahk(set(A),fun(fun(A,bool),bool)) = ball(A) ).

% INF_bool_eq
tff(fact_6407_INTER__eq,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5)) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_ahl(fun(B,set(A)),fun(set(B),fun(A,bool)),B4),A5)) ).

% INTER_eq
tff(fact_6408_Collect__ball__eq,axiom,
    ! [A: $tType,B: $tType,A5: set(B),P: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_ahm(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),A5),P)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aah(fun(A,fun(B,bool)),fun(B,set(A)),P)),A5)) ).

% Collect_ball_eq
tff(fact_6409_Ball__fold,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(A,bool,P,X4)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_ahn(fun(A,bool),fun(A,fun(bool,bool)),P),fTrue,A5)) ) ) ).

% Ball_fold
tff(fact_6410_Ball__Collect,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(A,bool,P,X4)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(fun(A,bool),set(A),collect(A),P))) ) ).

% Ball_Collect
tff(fact_6411_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: fun(B,set(C)),F3: fun(C,bool),G3: fun(A,B),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,aa(fun(C,bool),fun(B,bool),aTP_Lamp_aho(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),A5),F3)),G3),X5))
    <=> ! [Xa: C] :
          ( pp(aa(set(C),bool,member(C,Xa),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),comp(B,set(C),A,A5),G3),X5)))
         => pp(aa(C,bool,F3,Xa)) ) ) ).

% Ball_comp_iff
tff(fact_6412_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A5) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ahp(set(A),fun(A,bool),A5))) ) ).

% Inf_eq_Sup
tff(fact_6413_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A5) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ahq(set(A),fun(A,bool),A5))) ) ).

% Sup_eq_Inf
tff(fact_6414_takeWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys2: list(A)] :
      ( ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X3)) )
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys2)) ) )
      & ( ~ ! [X5: A] :
              ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X5)) )
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = takeWhile(A,P,Xs) ) ) ) ).

% takeWhile_append
tff(fact_6415_dropWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys2: list(A)] :
      ( ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X3)) )
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = dropWhile(A,P,Ys2) ) )
      & ( ~ ! [X5: A] :
              ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X5)) )
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys2) ) ) ) ).

% dropWhile_append
tff(fact_6416_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys2))))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),fequal(A)),X4)) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_6417_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ! [X3: product_prod(list(A),list(A))] :
          ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X3),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys2))))
         => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_ahr(list(A),fun(list(A),bool))),X3)) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ( concat(A,Xs) = concat(A,Ys2) )
        <=> ( Xs = Ys2 ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_6418_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys2) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ! [X3: product_prod(list(A),list(A))] :
              ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X3),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys2))))
             => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_ahr(list(A),fun(list(A),bool))),X3)) )
         => ( Xs = Ys2 ) ) ) ) ).

% concat_injective
tff(fact_6419_sorted__wrt_Opelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Y: bool] :
      ( ( sorted_wrt(A,X,Xa2)
      <=> pp(Y) )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa2)))
       => ( ( ( Xa2 = nil(A) )
           => ( pp(Y)
             => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),nil(A)))) ) )
         => ~ ! [X3: A,Ys3: list(A)] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
               => ( ( pp(Y)
                  <=> ( ! [Xa: A] :
                          ( pp(aa(set(A),bool,member(A,Xa),aa(list(A),set(A),set2(A),Ys3)))
                         => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa)) )
                      & sorted_wrt(A,X,Ys3) ) )
                 => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3)))) ) ) ) ) ) ).

% sorted_wrt.pelims(1)
tff(fact_6420_sorted__wrt_Opelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A)] :
      ( sorted_wrt(A,X,Xa2)
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa2)))
       => ( ( ( Xa2 = nil(A) )
           => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),nil(A)))) )
         => ~ ! [X3: A,Ys3: list(A)] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
               => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3))))
                 => ~ ( ! [Xa3: A] :
                          ( pp(aa(set(A),bool,member(A,Xa3),aa(list(A),set(A),set2(A),Ys3)))
                         => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa3)) )
                      & sorted_wrt(A,X,Ys3) ) ) ) ) ) ) ).

% sorted_wrt.pelims(2)
tff(fact_6421_Inter__eq,axiom,
    ! [A: $tType,A5: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ahs(set(set(A)),fun(A,bool),A5)) ).

% Inter_eq
tff(fact_6422_Sup__int__def,axiom,
    ! [X6: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X6) = the(int,aTP_Lamp_aht(set(int),fun(int,bool),X6)) ).

% Sup_int_def
tff(fact_6423_Union__maximal__sets,axiom,
    ! [A: $tType,F13: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F13))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ahu(set(set(A)),fun(set(A),bool),F13))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F13) ) ) ).

% Union_maximal_sets
tff(fact_6424_Sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),A5)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ahv(set(set(A)),fun(set(A),bool),A5)))) ) ).

% Sup_Inf
tff(fact_6425_Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A5)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ahv(set(set(A)),fun(set(A),bool),A5)))) ) ).

% Inf_Sup
tff(fact_6426_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),bool),set(filter(A)),collect(filter(A)),aTP_Lamp_ahw(set(filter(A)),fun(filter(A),bool),S))) ).

% Inf_filter_def
tff(fact_6427_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ahx(set(set(A)),fun(set(A),bool),A5))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A5)))) ) ).

% Sup_Inf_le
tff(fact_6428_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ahv(set(set(A)),fun(set(A),bool),A5)))))) ) ).

% Inf_Sup_le
tff(fact_6429_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A5: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ahy(set(set(A)),fun(set(A),bool),A5)))))) ) ).

% finite_Inf_Sup
tff(fact_6430_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G3: fun(B,A),A5: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_ahz(fun(B,A),fun(set(B),A),G3)),A5)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_aia(fun(B,A),fun(set(B),A),G3)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_aib(set(set(B)),fun(set(B),bool),A5)))) ) ).

% SUP_INF_set
tff(fact_6431_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G3: fun(B,A),A5: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_aia(fun(B,A),fun(set(B),A),G3)),A5)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_ahz(fun(B,A),fun(set(B),A),G3)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_aib(set(set(B)),fun(set(B),bool),A5)))) ) ).

% INF_SUP_set
tff(fact_6432_option__Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(option(A)))] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),complete_Sup_Sup(option(A))),A5))),aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),complete_Inf_Inf(option(A))),aa(fun(set(option(A)),bool),set(set(option(A))),collect(set(option(A))),aTP_Lamp_aic(set(set(option(A))),fun(set(option(A)),bool),A5)))))) ) ).

% option_Inf_Sup
tff(fact_6433_sorted__wrt_Opelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A)] :
      ( ~ sorted_wrt(A,X,Xa2)
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa2)))
       => ~ ! [X3: A,Ys3: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
             => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3))))
               => ( ! [Xa4: A] :
                      ( pp(aa(set(A),bool,member(A,Xa4),aa(list(A),set(A),set2(A),Ys3)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa4)) )
                  & sorted_wrt(A,X,Ys3) ) ) ) ) ) ).

% sorted_wrt.pelims(3)
tff(fact_6434_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] : pp(aa(fun(fun(A,bool),fun(A,bool)),bool,order_mono(fun(A,bool),fun(A,bool)),aTP_Lamp_aid(fun(A,A),fun(fun(A,bool),fun(A,bool)),F3))) ) ).

% iteratesp.mono
tff(fact_6435_times__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ra(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_ra(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% times_int.rsp
tff(fact_6436_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: nat] :
      ( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V2)))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_6437_zero__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat)))) ).

% zero_int.rsp
tff(fact_6438_chain__empty,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).

% chain_empty
tff(fact_6439_chain__compr,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),A5: set(A),P: fun(A,bool)] :
      ( comple1602240252501008431_chain(A,Ord,A5)
     => comple1602240252501008431_chain(A,Ord,aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),A5),P))) ) ).

% chain_compr
tff(fact_6440_ccpo__Sup__mono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A5: set(A),B4: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
         => ( comple1602240252501008431_chain(A,ord_less_eq(A),B4)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A5))
                 => ? [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),B4))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ) ) ).

% ccpo_Sup_mono
tff(fact_6441_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A5: set(A),X: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
         => ( pp(aa(set(A),bool,member(A,X),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ).

% ccpo_Sup_upper
tff(fact_6442_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A5: set(A),Z2: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z2)) ) ) ) ).

% ccpo_Sup_least
tff(fact_6443_chain__subset,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),A5: set(A),B4: set(A)] :
      ( comple1602240252501008431_chain(A,Ord,A5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => comple1602240252501008431_chain(A,Ord,B4) ) ) ).

% chain_subset
tff(fact_6444_uminus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_rb(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_rb(nat,fun(nat,product_prod(nat,nat)))))) ).

% uminus_int.rsp
tff(fact_6445_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_6446_nat_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),nat,nat,intrel,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)))) ).

% nat.rsp
tff(fact_6447_one__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat)))) ).

% one_int.rsp
tff(fact_6448_of__int_Orsp,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(fun(product_prod(nat,nat),A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(product_prod(nat,nat),A),bool),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_rc(nat,fun(nat,A)))),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rc(nat,fun(nat,A))))) ) ).

% of_int.rsp
tff(fact_6449_intrel__def,axiom,
    intrel = aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_aif(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).

% intrel_def
tff(fact_6450_less__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_re(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_re(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_int.rsp
tff(fact_6451_less__eq__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rg(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rg(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_eq_int.rsp
tff(fact_6452_minus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rk(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_rk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% minus_int.rsp
tff(fact_6453_plus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ri(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_ri(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% plus_int.rsp
tff(fact_6454_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A5: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
         => ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => pp(aa(set(A),bool,member(A,aa(set(A),A,complete_Sup_Sup(A),A5)),A5)) ) ) ) ) ).

% in_chain_finite
tff(fact_6455_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X5: fun(A,A)] : comple7512665784863727008ratesp(A,X5) = complete_lattice_lfp(fun(A,bool),aTP_Lamp_aid(fun(A,A),fun(fun(A,bool),fun(A,bool)),X5)) ) ).

% iteratesp_def
tff(fact_6456_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M6: set(A),F3: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M6)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),M6))
               => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X3)) )
           => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),aa(set(A),A,complete_Sup_Sup(A),M6))) ) ) ) ).

% iteratesp.Sup
tff(fact_6457_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A3: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),A3))
         => ( ! [X3: A] :
                ( ( A3 = aa(A,A,F3,X3) )
               => ~ pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X3)) )
           => ~ ! [M11: set(A)] :
                  ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M11) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M11)
                   => ~ ! [X5: A] :
                          ( pp(aa(set(A),bool,member(A,X5),M11))
                         => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X5)) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_6458_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A3: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),A3))
        <=> ( ? [X4: A] :
                ( ( A3 = aa(A,A,F3,X4) )
                & pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X4)) )
            | ? [M13: set(A)] :
                ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M13) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M13)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),M13))
                   => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X4)) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_6459_flat__lub__def,axiom,
    ! [A: $tType,A5: set(A),B2: A] :
      ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A5) = B2 ) )
      & ( ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A5) = the(A,aa(A,fun(A,bool),aTP_Lamp_aig(set(A),fun(A,fun(A,bool)),A5),B2)) ) ) ) ).

% flat_lub_def
tff(fact_6460_listrel__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : listrel(A,B,X5) = aa(fun(product_prod(list(A),list(B)),bool),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),bool)),fun(product_prod(list(A),list(B)),bool),product_case_prod(list(A),list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)))) ).

% listrel_def
tff(fact_6461_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),X5: list(A),Xa3: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),X5),Xa3))
    <=> pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X5),Xa3)),listrel(A,B,R3))) ) ).

% listrelp_listrel_eq
tff(fact_6462_listrel1__subset__listrel,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R6))
     => ( refl_on(A,top_top(set(A)),R6)
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel(A,A,R6))) ) ) ).

% listrel1_subset_listrel
tff(fact_6463_min__ext__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : min_ext(A,R3) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_aih(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R3)) ).

% min_ext_def
tff(fact_6464_bex__empty,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ~ ? [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),bot_bot(set(A))))
          & pp(aa(A,bool,P,X5)) ) ).

% bex_empty
tff(fact_6465_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A5: set(A),Q: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aii(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),Q))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),X4)))) ) ) ) ).

% finite_Collect_bex
tff(fact_6466_bex__UNIV,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),top_top(set(A))))
          & pp(aa(A,bool,P,X4)) )
    <=> ? [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).

% bex_UNIV
tff(fact_6467_Image__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,bool)),A5: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P))),A5) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_aij(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),P),A5)) ).

% Image_Collect_case_prod
tff(fact_6468_SUP__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_aik(set(A),fun(fun(A,bool),bool)) = bex(A) ).

% SUP_bool_eq
tff(fact_6469_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(A),set(B),image2(A,B,F3),A5)) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ail(fun(A,B),fun(set(A),fun(A,bool)),F3),A5)) ).

% vimage_image_eq
tff(fact_6470_Collect__bex__eq,axiom,
    ! [A: $tType,B: $tType,A5: set(B),P: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_aim(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),A5),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aah(fun(A,fun(B,bool)),fun(B,set(A)),P)),A5)) ).

% Collect_bex_eq
tff(fact_6471_UNION__eq,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),A5)) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_ain(fun(B,set(A)),fun(set(B),fun(A,bool)),B4),A5)) ).

% UNION_eq
tff(fact_6472_image__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] : aa(set(A),set(B),image2(A,B,F3),A5) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_aio(fun(A,B),fun(set(A),fun(B,bool)),F3),A5)) ).

% image_def
tff(fact_6473_Union__eq,axiom,
    ! [A: $tType,A5: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aip(set(set(A)),fun(A,bool),A5)) ).

% Union_eq
tff(fact_6474_Image__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(A)] : aa(set(A),set(B),image(A,B,R3),S2) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_aiq(set(product_prod(A,B)),fun(set(A),fun(B,bool)),R3),S2)) ).

% Image_def
tff(fact_6475_refl__on__Un,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),B4: set(A),S2: set(product_prod(A,A))] :
      ( refl_on(A,A5,R3)
     => ( refl_on(A,B4,S2)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4),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))),R3),S2)) ) ) ).

% refl_on_Un
tff(fact_6476_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_6477_refl__onD,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,A5,R3)
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R3)) ) ) ).

% refl_onD
tff(fact_6478_refl__onD1,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A5,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => pp(aa(set(A),bool,member(A,X),A5)) ) ) ).

% refl_onD1
tff(fact_6479_refl__onD2,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A5,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => pp(aa(set(A),bool,member(A,Y),A5)) ) ) ).

% refl_onD2
tff(fact_6480_refl__on__Id__on,axiom,
    ! [A: $tType,A5: set(A)] : refl_on(A,A5,id_on(A,A5)) ).

% refl_on_Id_on
tff(fact_6481_refl__Id,axiom,
    ! [A: $tType] : refl_on(A,top_top(set(A)),id2(A)) ).

% refl_Id
tff(fact_6482_refl__on__Int,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),B4: set(A),S2: set(product_prod(A,A))] :
      ( refl_on(A,A5,R3)
     => ( refl_on(A,B4,S2)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),S2)) ) ) ).

% refl_on_Int
tff(fact_6483_Bex__fold,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
            & pp(aa(A,bool,P,X4)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_air(fun(A,bool),fun(A,fun(bool,bool)),P),fFalse,A5)) ) ) ).

% Bex_fold
tff(fact_6484_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat),B4: set(nat)] : nths(A,nths(A,Xs,A5),B4) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ait(set(nat),fun(set(nat),fun(nat,bool)),A5),B4))) ).

% nths_nths
tff(fact_6485_refl__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : refl_on(A,top_top(set(A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),id2(A))) ).

% refl_reflcl
tff(fact_6486_refl__onI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3)) )
       => refl_on(A,A5,R3) ) ) ).

% refl_onI
tff(fact_6487_refl__on__def,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A5,R3)
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R3)) ) ) ) ).

% refl_on_def
tff(fact_6488_max__extp_Ocases,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),A1: set(A),A22: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R2),A1),A22))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
         => ( pp(aa(set(A),bool,finite_finite2(A),A22))
           => ( ( A22 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
             => ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),A1))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),A22))
                        & pp(aa(A,bool,aa(A,fun(A,bool),R2,X5),Xa4)) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_6489_max__extp_Osimps,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),A1: set(A),A22: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R2),A1),A22))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
        & pp(aa(set(A),bool,finite_finite2(A),A22))
        & ( A22 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A1))
           => ? [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),A22))
                & pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),Xa)) ) ) ) ) ).

% max_extp.simps
tff(fact_6490_max__extp_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),R2: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
       => ( ( Y6 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),Y6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Xa3)) ) )
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R2),X6),Y6)) ) ) ) ) ).

% max_extp.max_extI
tff(fact_6491_refl__on__def_H,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A5,R3)
    <=> ( ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_aiu(set(A),fun(A,fun(A,bool)),A5)),X4)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R3)) ) ) ) ).

% refl_on_def'
tff(fact_6492_refl__on__reflcl__Image,axiom,
    ! [A: $tType,B4: set(A),A5: set(product_prod(A,A)),C4: set(A)] :
      ( refl_on(A,B4,A5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B4))
       => ( aa(set(A),set(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A5),id2(A))),C4) = aa(set(A),set(A),image(A,A,A5),C4) ) ) ) ).

% refl_on_reflcl_Image
tff(fact_6493_refl__on__UNION,axiom,
    ! [B: $tType,A: $tType,S: set(A),A5: fun(A,set(B)),R3: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),S))
         => refl_on(B,aa(A,set(B),A5,X3),aa(A,set(product_prod(B,B)),R3,X3)) )
     => refl_on(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),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)),R3),S))) ) ).

% refl_on_UNION
tff(fact_6494_refl__on__INTER,axiom,
    ! [B: $tType,A: $tType,S: set(A),A5: fun(A,set(B)),R3: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),S))
         => refl_on(B,aa(A,set(B),A5,X3),aa(A,set(product_prod(B,B)),R3,X3)) )
     => refl_on(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),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)),R3),S))) ) ).

% refl_on_INTER
tff(fact_6495_refl__on__singleton,axiom,
    ! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_6496_map__project__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),A5: set(A)] : map_project(A,B,F3,A5) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_aiv(fun(A,option(B)),fun(set(A),fun(B,bool)),F3),A5)) ).

% map_project_def
tff(fact_6497_refl__on__domain,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,A5,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( pp(aa(set(A),bool,member(A,A3),A5))
          & pp(aa(set(A),bool,member(A,B2),A5)) ) ) ) ).

% refl_on_domain
tff(fact_6498_linear__order__on__singleton,axiom,
    ! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_6499_max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : max_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_aiw(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R2))) ).

% max_ext_eq
tff(fact_6500_ball__UNIV,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),top_top(set(A))))
         => pp(aa(A,bool,P,X4)) )
    <=> ! [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).

% ball_UNIV
tff(fact_6501_rel__fun__def,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A5: fun(A,fun(C,bool)),B4: fun(B,fun(D,bool)),X5: fun(A,B),Xa3: fun(C,D)] :
      ( pp(aa(fun(C,D),bool,aa(fun(A,B),fun(fun(C,D),bool),bNF_rel_fun(A,C,B,D,A5,B4),X5),Xa3))
    <=> ! [Xb5: A,Y5: C] :
          ( pp(aa(C,bool,aa(A,fun(C,bool),A5,Xb5),Y5))
         => pp(aa(D,bool,aa(B,fun(D,bool),B4,aa(A,B,X5,Xb5)),aa(C,D,Xa3,Y5))) ) ) ).

% rel_fun_def
tff(fact_6502_rel__fun__eq__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: fun(B,fun(C,bool)),X5: fun(A,B),Xa3: fun(A,C)] :
      ( pp(aa(fun(A,C),bool,aa(fun(A,B),fun(fun(A,C),bool),bNF_rel_fun(A,A,B,C,fequal(A),R2),X5),Xa3))
    <=> ! [Xb5: A] : pp(aa(C,bool,aa(B,fun(C,bool),R2,aa(A,B,X5,Xb5)),aa(A,C,Xa3,Xb5))) ) ).

% rel_fun_eq_rel
tff(fact_6503_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_6504_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),D3: B] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_aix(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),A5),B4),D3)))) ) ) ).

% finite_set_of_finite_funs
tff(fact_6505_Collect__all__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aiy(fun(A,fun(B,bool)),fun(A,bool),P)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aah(fun(A,fun(B,bool)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_all_eq
tff(fact_6506_Func__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] : bNF_Wellorder_Func(A,B,A5,B4) = aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aa(set(B),fun(fun(A,B),bool),aTP_Lamp_aiz(set(A),fun(set(B),fun(fun(A,B),bool)),A5),B4)) ).

% Func_def
tff(fact_6507_linear__order__on__Restr,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A] :
      ( order_679001287576687338der_on(A,A5,R3)
     => order_679001287576687338der_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),order_above(A,R3,X)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_above(A,R3,X),aa(A,fun(A,set(A)),aTP_Lamp_aja(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X)))) ) ).

% linear_order_on_Restr
tff(fact_6508_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_ajb(fun(A,bool),fun(A,bool),P)) ) ).

% Greatest_def
tff(fact_6509_GreatestI__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_nat
tff(fact_6510_Greatest__le__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),order_Greatest(nat,P))) ) ) ).

% Greatest_le_nat
tff(fact_6511_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,bool),B2: nat] :
      ( ? [X_13: nat] : pp(aa(nat,bool,P,X_13))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_ex_nat
tff(fact_6512_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y4: A] :
                        ( pp(aa(A,bool,P,Y4))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,order_Greatest(A,P))) ) ) ) ) ).

% GreatestI2_order
tff(fact_6513_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_6514_above__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : order_above(A,R3,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),A3)) ).

% above_def
tff(fact_6515_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),P: fun(A,bool)] : aa(fun(A,bool),A,least(A,Less_eq),P) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajc(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P)) ).

% ord.Least_def
tff(fact_6516_transfer__bforall__def,axiom,
    ! [A: $tType,X5: fun(A,bool),Xa3: fun(A,bool)] :
      ( transfer_bforall(A,X5,Xa3)
    <=> ! [Xb5: A] :
          ( pp(aa(A,bool,X5,Xb5))
         => pp(aa(A,bool,Xa3,Xb5)) ) ) ).

% transfer_bforall_def
tff(fact_6517_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_6518_subset__mset_OLeast__def,axiom,
    ! [A: $tType,P: fun(multiset(A),bool)] : aa(fun(multiset(A),bool),multiset(A),least(multiset(A),subseteq_mset(A)),P) = the(multiset(A),aTP_Lamp_ajd(fun(multiset(A),bool),fun(multiset(A),bool),P)) ).

% subset_mset.Least_def
tff(fact_6519_Total__subset__Id,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),id2(A)))
       => ( ( R3 = bot_bot(set(product_prod(A,A))) )
          | ? [A6: A] : R3 = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),A6)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_6520_lists__empty,axiom,
    ! [A: $tType] : lists(A,bot_bot(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% lists_empty
tff(fact_6521_lists__Int__eq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,A5)),lists(A,B4)) ).

% lists_Int_eq
tff(fact_6522_Field__square,axiom,
    ! [A: $tType,X: set(A)] : aa(set(product_prod(A,A)),set(A),field2(A),product_Sigma(A,A,X,aTP_Lamp_yj(set(A),fun(A,set(A)),X))) = X ).

% Field_square
tff(fact_6523_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_6524_finite__Field__eq__finite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2)))
    <=> pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2)) ) ).

% finite_Field_eq_finite
tff(fact_6525_Field__Un,axiom,
    ! [A: $tType,R3: 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))),R3),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),R3)),aa(set(product_prod(A,A)),set(A),field2(A),S2)) ).

% Field_Un
tff(fact_6526_Field__Union,axiom,
    ! [A: $tType,R2: 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))),R2)) = 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)),R2)) ).

% Field_Union
tff(fact_6527_Field__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R3: 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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R3)) ).

% Field_insert
tff(fact_6528_lists__IntI,axiom,
    ! [A: $tType,L: list(A),A5: set(A),B4: set(A)] :
      ( pp(aa(set(list(A)),bool,member(list(A),L),lists(A,A5)))
     => ( pp(aa(set(list(A)),bool,member(list(A),L),lists(A,B4)))
       => pp(aa(set(list(A)),bool,member(list(A),L),lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))) ) ) ).

% lists_IntI
tff(fact_6529_FieldI2,axiom,
    ! [A: $tType,I2: A,J: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J)),R2))
     => pp(aa(set(A),bool,member(A,J),aa(set(product_prod(A,A)),set(A),field2(A),R2))) ) ).

% FieldI2
tff(fact_6530_FieldI1,axiom,
    ! [A: $tType,I2: A,J: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J)),R2))
     => pp(aa(set(A),bool,member(A,I2),aa(set(product_prod(A,A)),set(A),field2(A),R2))) ) ).

% FieldI1
tff(fact_6531_mono__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),R3)),aa(set(product_prod(A,A)),set(A),field2(A),S2))) ) ).

% mono_Field
tff(fact_6532_lists__image__witness,axiom,
    ! [A: $tType,B: $tType,X: list(A),F3: fun(B,A),Q: set(B)] :
      ( pp(aa(set(list(A)),bool,member(list(A),X),lists(A,aa(set(B),set(A),image2(B,A,F3),Q))))
     => ~ ! [Xo2: list(B)] :
            ( pp(aa(set(list(B)),bool,member(list(B),Xo2),lists(B,Q)))
           => ( X != aa(list(B),list(A),map(B,A,F3),Xo2) ) ) ) ).

% lists_image_witness
tff(fact_6533_finite__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R3))
     => pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ).

% finite_Field
tff(fact_6534_R__subset__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_aje(set(product_prod(A,A)),fun(A,set(A)),R2)))) ).

% R_subset_Field
tff(fact_6535_Restr__Field,axiom,
    ! [A: $tType,R3: 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))),R3),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_aje(set(product_prod(A,A)),fun(A,set(A)),R3))) = R3 ).

% Restr_Field
tff(fact_6536_lists__eq__set,axiom,
    ! [A: $tType,A5: set(A)] : lists(A,A5) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_ajf(set(A),fun(list(A),bool),A5)) ).

% lists_eq_set
tff(fact_6537_Field__not__elem,axiom,
    ! [A: $tType,V2: A,R2: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,member(A,V2),aa(set(product_prod(A,A)),set(A),field2(A),R2)))
     => ! [X5: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),R2))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ajg(A,fun(A,fun(A,bool)),V2)),X5)) ) ) ).

% Field_not_elem
tff(fact_6538_fst__in__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_fst(A,A)),R2)),aa(set(product_prod(A,A)),set(A),field2(A),R2))) ).

% fst_in_Field
tff(fact_6539_snd__in__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_snd(A,A)),R2)),aa(set(product_prod(A,A)),set(A),field2(A),R2))) ).

% snd_in_Field
tff(fact_6540_lists__mono,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),lists(A,A5)),lists(A,B4))) ) ).

% lists_mono
tff(fact_6541_rel__restrict__Int__empty,axiom,
    ! [A: $tType,A5: 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)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R2)) = bot_bot(set(A)) )
     => ( rel_restrict(A,R2,A5) = R2 ) ) ).

% rel_restrict_Int_empty
tff(fact_6542_Field__rel__restrict,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),rel_restrict(A,R2,A5))),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),R2)),A5))) ).

% Field_rel_restrict
tff(fact_6543_Field__Restr__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))),A5)) ).

% Field_Restr_subset
tff(fact_6544_trancl__subset__Field2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_aje(set(product_prod(A,A)),fun(A,set(A)),R3)))) ).

% trancl_subset_Field2
tff(fact_6545_Field__natLeq__on,axiom,
    ! [N2: nat] : aa(set(product_prod(nat,nat)),set(nat),field2(nat),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ajh(nat,fun(nat,fun(nat,bool)),N2)))) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2)) ).

% Field_natLeq_on
tff(fact_6546_Refl__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => refl_on(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% Refl_Restr
tff(fact_6547_Total__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => total_on(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% Total_Restr
tff(fact_6548_total__on__imp__Total__Restr,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( total_on(A,A5,R3)
     => total_on(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% total_on_imp_Total_Restr
tff(fact_6549_Linear__order__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => order_679001287576687338der_on(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% Linear_order_Restr
tff(fact_6550_lists__of__len__fin1,axiom,
    ! [A: $tType,P: set(A),N2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),P))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_aji(nat,fun(list(A),bool),N2))))) ) ).

% lists_of_len_fin1
tff(fact_6551_lists__of__len__fin2,axiom,
    ! [A: $tType,P: set(A),N2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),P))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_ajj(nat,fun(list(A),bool),N2))))) ) ).

% lists_of_len_fin2
tff(fact_6552_rtrancl__Image__in__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),V: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),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),R2)),V))) ).

% rtrancl_Image_in_Field
tff(fact_6553_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R3),id2(A)))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_6554_Total__Id__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),id2(A)))
       => ( aa(set(product_prod(A,A)),set(A),field2(A),R3) = 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))),minus_minus(set(product_prod(A,A))),R3),id2(A))) ) ) ) ).

% Total_Id_Field
tff(fact_6555_Refl__Field__Restr2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( 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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) = A5 ) ) ) ).

% Refl_Field_Restr2
tff(fact_6556_Refl__Field__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( 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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) = 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),field2(A),R3)),A5) ) ) ).

% Refl_Field_Restr
tff(fact_6557_listrel__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),product_Sigma(list(A),list(A),lists(A,A5),aTP_Lamp_ajk(set(A),fun(list(A),set(list(A))),A5)))) ) ).

% listrel_subset
tff(fact_6558_UnderS__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] : order_UnderS(A,R3,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ajl(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A5)) ).

% UnderS_def
tff(fact_6559_Under__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] : order_Under(A,R3,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ajm(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A5)) ).

% Under_def
tff(fact_6560_Above__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] : order_Above(A,R3,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ajn(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A5)) ).

% Above_def
tff(fact_6561_subset__mset_OGreatest__def,axiom,
    ! [A: $tType,P: fun(multiset(A),bool)] : aa(fun(multiset(A),bool),multiset(A),greatest(multiset(A),subseteq_mset(A)),P) = the(multiset(A),aTP_Lamp_ajo(fun(multiset(A),bool),fun(multiset(A),bool),P)) ).

% subset_mset.Greatest_def
tff(fact_6562_order_OGreatest_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : greatest(A,Less_eq) = greatest(A,Less_eq) ).

% order.Greatest.cong
tff(fact_6563_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3)) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_6564_bsqr__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R3) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_ajq(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R3)))) ).

% bsqr_def
tff(fact_6565_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_6566_Field__bsqr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : aa(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)),field2(product_prod(A,A)),bNF_Wellorder_bsqr(A,R3)) = product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_aje(set(product_prod(A,A)),fun(A,set(A)),R3)) ).

% Field_bsqr
tff(fact_6567_Preorder__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => order_preorder_on(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% Preorder_Restr
tff(fact_6568_subset__Image__Image__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),A5)),aa(set(A),set(A),image(A,A,R3),B4)))
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),A5))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),B4))
                    & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X4)),R3)) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_6569_cofinal__def,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A5,R3)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ? [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),A5))
              & ( X4 != Xa )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R3)) ) ) ) ).

% cofinal_def
tff(fact_6570_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( 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))),R3),id2(A)))
      <=> ! [A13: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A13),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
           => ( ( A13 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A13))
                  & ! [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),A13))
                     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R3)) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_6571_wf__empty,axiom,
    ! [A: $tType] : wf(A,bot_bot(set(product_prod(A,A)))) ).

% wf_empty
tff(fact_6572_brk__rel__wf,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(product_prod(bool,A),brk_rel(A,A,R2)) ) ).

% brk_rel_wf
tff(fact_6573_wf__insert,axiom,
    ! [A: $tType,Y: A,X: A,R3: 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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R3))
    <=> ( wf(A,R3)
        & ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3))) ) ) ).

% wf_insert
tff(fact_6574_wf__Int2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( wf(A,R3)
     => 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))),R6),R3)) ) ).

% wf_Int2
tff(fact_6575_wf__Int1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( wf(A,R3)
     => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),R6)) ) ).

% wf_Int1
tff(fact_6576_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,bool),F3: fun(A,nat),G3: fun(A,A)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,aa(A,A,G3,X3))),aa(A,nat,F3,X3))) )
     => wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_ajr(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),P),G3)))) ) ).

% wf_if_measure
tff(fact_6577_wf__subset__mset__rel,axiom,
    ! [A: $tType] : wf(multiset(A),aa(fun(product_prod(multiset(A),multiset(A)),bool),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),bool)),fun(product_prod(multiset(A),multiset(A)),bool),product_case_prod(multiset(A),multiset(A),bool),subset_mset(A)))) ).

% wf_subset_mset_rel
tff(fact_6578_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))) ) ).

% wf
tff(fact_6579_wf__no__loop,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( ( relcomp(B,B,B,R2,R2) = bot_bot(set(product_prod(B,B))) )
     => wf(B,R2) ) ).

% wf_no_loop
tff(fact_6580_wfE__min_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R2)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z3: A] :
              ( pp(aa(set(A),bool,member(A,Z3),Q))
             => ~ ! [Y4: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3)),R2))
                   => ~ pp(aa(set(A),bool,member(A,Y4),Q)) ) ) ) ) ).

% wfE_min'
tff(fact_6581_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ~ ? [F10: fun(nat,A)] :
          ! [I4: nat] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F10,aa(nat,nat,suc,I4))),aa(nat,A,F10,I4))),R3)) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_6582_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),F3: fun(nat,A)] :
      ( wf(A,R3)
     => ~ ! [K2: nat] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F3,aa(nat,nat,suc,K2))),aa(nat,A,F3,K2))),R3)) ) ).

% wf_no_infinite_down_chainE
tff(fact_6583_wf__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ! [P8: fun(A,bool)] :
          ( ! [X4: A] :
              ( ! [Y5: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4)),R3))
                 => pp(aa(A,bool,P8,Y5)) )
             => pp(aa(A,bool,P8,X4)) )
         => ! [X_12: A] : pp(aa(A,bool,P8,X_12)) ) ) ).

% wf_def
tff(fact_6584_wfE__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Q: set(A)] :
      ( wf(A,R2)
     => ( pp(aa(set(A),bool,member(A,X),Q))
       => ~ ! [Z3: A] :
              ( pp(aa(set(A),bool,member(A,Z3),Q))
             => ~ ! [Y4: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3)),R2))
                   => ~ pp(aa(set(A),bool,member(A,Y4),Q)) ) ) ) ) ).

% wfE_min
tff(fact_6585_wfI__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X3: A,Q2: set(A)] :
          ( pp(aa(set(A),bool,member(A,X3),Q2))
         => ? [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),Q2))
              & ! [Y3: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa3)),R2))
                 => ~ pp(aa(set(A),bool,member(A,Y3),Q2)) ) ) )
     => wf(A,R2) ) ).

% wfI_min
tff(fact_6586_wfUNIVI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ! [P2: fun(A,bool),X3: A] :
          ( ! [Xa3: A] :
              ( ! [Y3: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa3)),R3))
                 => pp(aa(A,bool,P2,Y3)) )
             => pp(aa(A,bool,P2,Xa3)) )
         => pp(aa(A,bool,P2,X3)) )
     => wf(A,R3) ) ).

% wfUNIVI
tff(fact_6587_wf__asym,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R3))
       => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R3)) ) ) ).

% wf_asym
tff(fact_6588_wf__induct,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( wf(A,R3)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R3))
               => pp(aa(A,bool,P,Y4)) )
           => pp(aa(A,bool,P,X3)) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% wf_induct
tff(fact_6589_wf__irrefl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( wf(A,R3)
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R3)) ) ).

% wf_irrefl
tff(fact_6590_wf__not__sym,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R3))
       => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R3)) ) ) ).

% wf_not_sym
tff(fact_6591_wf__not__refl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( wf(A,R3)
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R3)) ) ).

% wf_not_refl
tff(fact_6592_wf__eq__minimal,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ! [Q9: set(A)] :
          ( ? [X4: A] : pp(aa(set(A),bool,member(A,X4),Q9))
         => ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Q9))
              & ! [Y5: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4)),R3))
                 => ~ pp(aa(set(A),bool,member(A,Y5),Q9)) ) ) ) ) ).

% wf_eq_minimal
tff(fact_6593_wf__induct__rule,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( wf(A,R3)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R3))
               => pp(aa(A,bool,P,Y4)) )
           => pp(aa(A,bool,P,X3)) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% wf_induct_rule
tff(fact_6594_wf__union__merge,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S))
    <=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),relcomp(A,A,A,R2,R2)),relcomp(A,A,A,S,R2))),S)) ) ).

% wf_union_merge
tff(fact_6595_wf__less,axiom,
    wf(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat)))) ).

% wf_less
tff(fact_6596_wf__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(A,A))] :
      ( wf(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),P3),R3))
       => wf(A,P3) ) ) ).

% wf_subset
tff(fact_6597_wf__relcomp__compatible,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),relcomp(A,A,A,S,R2)))
       => wf(A,relcomp(A,A,A,S,R2)) ) ) ).

% wf_relcomp_compatible
tff(fact_6598_wf__bounded__measure,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Ub: fun(A,nat),F3: fun(A,nat)] :
      ( ! [A6: A,B5: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6)),R3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B5)),aa(A,nat,Ub,A6)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,B5)),aa(A,nat,Ub,A6)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,A6)),aa(A,nat,F3,B5))) ) )
     => wf(A,R3) ) ).

% wf_bounded_measure
tff(fact_6599_wfE__pf,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] :
      ( wf(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),image(A,A,R2),A5)))
       => ( A5 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_6600_wfI__pf,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [A11: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),aa(set(A),set(A),image(A,A,R2),A11)))
         => ( A11 = bot_bot(set(A)) ) )
     => wf(A,R2) ) ).

% wfI_pf
tff(fact_6601_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),P: fun(B,bool),K: B,M: fun(B,A)] :
      ( wf(A,R3)
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),transitive_trancl(A,R3)))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),transitive_rtrancl(A,R3))) )
       => ( pp(aa(B,bool,P,K))
         => ? [X3: B] :
              ( pp(aa(B,bool,P,X3))
              & ! [Y4: B] :
                  ( pp(aa(B,bool,P,Y4))
                 => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M,X3)),aa(B,A,M,Y4))),transitive_rtrancl(A,R3))) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_6602_wf__union__compatible,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( wf(A,S)
       => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),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))),sup_sup(set(product_prod(A,A))),R2),S)) ) ) ) ).

% wf_union_compatible
tff(fact_6603_wfI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))))
     => ( ! [X3: A,P2: fun(A,bool)] :
            ( ! [Xa3: A] :
                ( ! [Y3: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa3)),R3))
                   => pp(aa(A,bool,P2,Y3)) )
               => pp(aa(A,bool,P2,Xa3)) )
           => ( pp(aa(set(A),bool,member(A,X3),A5))
             => ( pp(aa(set(A),bool,member(A,X3),B4))
               => pp(aa(A,bool,P2,X3)) ) ) )
       => wf(A,R3) ) ) ).

% wfI
tff(fact_6604_wf__eq__minimal2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ! [A13: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A13),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
            & ( A13 != bot_bot(set(A)) ) )
         => ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A13))
              & ! [Xa: A] :
                  ( pp(aa(set(A),bool,member(A,Xa),A13))
                 => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X4)),R3)) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_6605_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),Ub: fun(A,set(B)),F3: fun(A,set(B))] :
      ( ! [A6: A,B5: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6)),R3))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Ub,A6)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B5)),aa(A,set(B),Ub,A6)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,B5)),aa(A,set(B),Ub,A6)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F3,A6)),aa(A,set(B),F3,B5))) ) )
     => wf(A,R3) ) ).

% wf_bounded_set
tff(fact_6606_qc__wf__relto__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),relcomp(A,A,A,transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S)),R2)))
     => ( wf(A,relcomp(A,A,A,transitive_rtrancl(A,S),relcomp(A,A,A,R2,transitive_rtrancl(A,S))))
      <=> wf(A,R2) ) ) ).

% qc_wf_relto_iff
tff(fact_6607_wf__bounded__supset,axiom,
    ! [A: $tType,S: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_ajs(set(A),fun(set(A),fun(set(A),bool)),S)))) ) ).

% wf_bounded_supset
tff(fact_6608_finite__subset__wf,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_ajt(set(A),fun(set(A),fun(set(A),bool)),A5)))) ) ).

% finite_subset_wf
tff(fact_6609_reduction__pairI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),R2))
       => 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))),R2),S)) ) ) ).

% reduction_pairI
tff(fact_6610_reduction__pair__lemma,axiom,
    ! [A: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( fun_reduction_pair(A,P)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P)))
       => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P)))
         => ( wf(A,S)
           => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S)) ) ) ) ) ).

% reduction_pair_lemma
tff(fact_6611_reduction__pair__def,axiom,
    ! [A: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A)))] :
      ( fun_reduction_pair(A,P)
    <=> ( wf(A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P))
        & pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P))) ) ) ).

% reduction_pair_def
tff(fact_6612_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,bool)))] :
      ( wf(A,R2)
     => ( ! [F2: fun(A,B),G2: fun(A,B),X3: A,R: B] :
            ( ! [Z5: A] :
                ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),X3)),R2))
               => ( aa(A,B,F2,Z5) = aa(A,B,G2,Z5) ) )
           => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X3),R))
            <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,G2),X3),R)) ) )
       => ( ! [X3: A,F2: fun(A,B)] :
              ( ! [Y4: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R2))
                 => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),Y4),aa(A,B,F2,Y4))) )
             => ? [X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X3),X_13)) )
         => ? [F2: fun(A,B)] :
            ! [X5: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X5),aa(A,B,F2,X5))) ) ) ) ).

% dependent_wf_choice
tff(fact_6613_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( antisym(A,R3)
       => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
           => ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) )
            <=> ( A3 = B2 ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_6614_antisym__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),id2(A)))
    <=> antisym(A,R3) ) ).

% antisym_reflcl
tff(fact_6615_antisymD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( antisym(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3))
         => ( A3 = B2 ) ) ) ) ).

% antisymD
tff(fact_6616_antisymI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),R3))
           => ( X3 = Y3 ) ) )
     => antisym(A,R3) ) ).

% antisymI
tff(fact_6617_antisym__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( antisym(A,R3)
    <=> ! [X4: A,Y5: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4)),R3))
           => ( X4 = Y5 ) ) ) ) ).

% antisym_def
tff(fact_6618_antisym__Id,axiom,
    ! [A: $tType] : antisym(A,id2(A)) ).

% antisym_Id
tff(fact_6619_antisym__empty,axiom,
    ! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).

% antisym_empty
tff(fact_6620_antisym__Id__on,axiom,
    ! [A: $tType,A5: set(A)] : antisym(A,id_on(A,A5)) ).

% antisym_Id_on
tff(fact_6621_antisym__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => ( antisym(A,S2)
       => antisym(A,R3) ) ) ).

% antisym_subset
tff(fact_6622_antisym__singleton,axiom,
    ! [A: $tType,X: product_prod(A,A)] : antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),X),bot_bot(set(product_prod(A,A))))) ).

% antisym_singleton
tff(fact_6623_dependent__wellorder__choice,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(A)
     => ! [P: fun(fun(A,B),fun(A,fun(B,bool)))] :
          ( ! [R: B,F2: fun(A,B),G2: fun(A,B),X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3))
                 => ( aa(A,B,F2,Y4) = aa(A,B,G2,Y4) ) )
             => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X3),R))
              <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,G2),X3),R)) ) )
         => ( ! [X3: A,F2: fun(A,B)] :
                ( ! [Y4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3))
                   => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),Y4),aa(A,B,F2,Y4))) )
               => ? [X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X3),X_13)) )
           => ? [F2: fun(A,B)] :
              ! [X5: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X5),aa(A,B,F2,X5))) ) ) ) ).

% dependent_wellorder_choice
tff(fact_6624_antisym__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( antisym(A,R3)
     => antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% antisym_Restr
tff(fact_6625_chains__extend,axiom,
    ! [A: $tType,C2: set(set(A)),S: set(set(A)),Z2: set(A)] :
      ( pp(aa(set(set(set(A))),bool,member(set(set(A)),C2),chains2(A,S)))
     => ( pp(aa(set(set(A)),bool,member(set(A),Z2),S))
       => ( ! [X3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X3),C2))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Z2)) )
         => pp(aa(set(set(set(A))),bool,member(set(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Z2),bot_bot(set(set(A))))),C2)),chains2(A,S))) ) ) ) ).

% chains_extend
tff(fact_6626_rp__inv__image__rp,axiom,
    ! [A: $tType,B: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),F3: fun(B,A)] :
      ( fun_reduction_pair(A,P)
     => fun_reduction_pair(B,aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),fun_rp_inv_image(A,B),P),F3)) ) ).

% rp_inv_image_rp
tff(fact_6627_chainsD,axiom,
    ! [A: $tType,C2: set(set(A)),S: set(set(A)),X: set(A),Y: set(A)] :
      ( pp(aa(set(set(set(A))),bool,member(set(set(A)),C2),chains2(A,S)))
     => ( pp(aa(set(set(A)),bool,member(set(A),X),C2))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y),C2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X),Y))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y),X)) ) ) ) ) ).

% chainsD
tff(fact_6628_Zorn__Lemma2,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,member(set(set(A)),X3),chains2(A,A5)))
         => ? [Xa3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa3),A5))
              & ! [Xb3: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),Xb3),X3))
                 => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xb3),Xa3)) ) ) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A5))
          & ! [Xa3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa3),A5))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa3))
               => ( Xa3 = X3 ) ) ) ) ) ).

% Zorn_Lemma2
tff(fact_6629_chainsD2,axiom,
    ! [A: $tType,C2: set(set(A)),S: set(set(A))] :
      ( pp(aa(set(set(set(A))),bool,member(set(set(A)),C2),chains2(A,S)))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C2),S)) ) ).

% chainsD2
tff(fact_6630_Zorn__Lemma,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,member(set(set(A)),X3),chains2(A,A5)))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X3)),A5)) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A5))
          & ! [Xa3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa3),A5))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa3))
               => ( Xa3 = X3 ) ) ) ) ) ).

% Zorn_Lemma
tff(fact_6631_chains__def,axiom,
    ! [A: $tType,A5: set(set(A))] : chains2(A,A5) = aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_aju(set(set(A)),fun(set(set(A)),bool),A5)) ).

% chains_def
tff(fact_6632_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_ajv(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_6633_in__inv__image,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R3: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),inv_image(B,A,R3,F3)))
    <=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,X)),aa(A,B,F3,Y))),R3)) ) ).

% in_inv_image
tff(fact_6634_chain__subset__def,axiom,
    ! [A: $tType,C4: set(set(A))] :
      ( chain_subset(A,C4)
    <=> ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),C4))
         => ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa),C4))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa))
                | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa),X4)) ) ) ) ) ).

% chain_subset_def
tff(fact_6635_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(B,B)),F3: fun(A,B)] : inv_image(B,A,R3,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ajw(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),R3),F3))) ).

% inv_image_def
tff(fact_6636_total__inv__image,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),R3: set(product_prod(B,B))] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( total_on(B,top_top(set(B)),R3)
       => total_on(A,top_top(set(A)),inv_image(B,A,R3,F3)) ) ) ).

% total_inv_image
tff(fact_6637_lenlex__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R3)),aTP_Lamp_ajx(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_6638_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
         => order_antimono(nat,A,aTP_Lamp_ajy(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_6639_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),less_than))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ).

% less_than_iff
tff(fact_6640_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoD
tff(fact_6641_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoE
tff(fact_6642_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3))) )
         => order_antimono(A,B,F3) ) ) ).

% antimonoI
tff(fact_6643_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_antimono(A,B,F3)
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y5)),aa(A,B,F3,X4))) ) ) ) ).

% antimono_def
tff(fact_6644_antimono__iff__le__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_antimono(nat,A,F3)
        <=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N))),aa(nat,A,F3,N))) ) ) ).

% antimono_iff_le_Suc
tff(fact_6645_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F3,X)),aa(A,B,F3,Y)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) ) ) ) ).

% max_of_antimono
tff(fact_6646_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F3,X)),aa(A,B,F3,Y)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) ) ) ) ).

% min_of_antimono
tff(fact_6647_mlex__prod__def,axiom,
    ! [A: $tType,F3: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F3,R2) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R2),aTP_Lamp_ajz(fun(A,nat),fun(A,product_prod(nat,A)),F3)) ).

% mlex_prod_def
tff(fact_6648_sum__mset__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Y: B,A5: multiset(A)] : comm_m7189776963980413722m_mset(B,aa(multiset(A),multiset(B),image_mset(A,B,aTP_Lamp_aka(B,fun(A,B),Y)),A5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(multiset(A),nat,size_size(multiset(A)),A5))),Y) ) ).

% sum_mset_constant
tff(fact_6649_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z2: B,Y: B,A3: A] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A5),Y))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => ? [Y8: B] :
                ( ( Y = aa(B,B,aa(A,fun(B,B),F3,A3),Y8) )
                & pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),Y8)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_6650_Union__image__single__mset,axiom,
    ! [A: $tType,M: multiset(A)] : comm_m7189776963980413722m_mset(multiset(A),aa(multiset(A),multiset(multiset(A)),image_mset(A,multiset(A),aTP_Lamp_akb(A,multiset(A))),M)) = M ).

% Union_image_single_mset
tff(fact_6651_sum__mset_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_do(B,A)),A5)) = zero_zero(A) ) ).

% sum_mset.neutral_const
tff(fact_6652_fold__graph_Ocases,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
      ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A1),A22))
     => ( ( ( A1 = bot_bot(set(A)) )
         => ( A22 != Z2 ) )
       => ~ ! [X3: A,A11: set(A)] :
              ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),A11) )
             => ! [Y3: B] :
                  ( ( A22 = aa(B,B,aa(A,fun(B,B),F3,X3),Y3) )
                 => ( ~ pp(aa(set(A),bool,member(A,X3),A11))
                   => ~ pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A11),Y3)) ) ) ) ) ) ).

% fold_graph.cases
tff(fact_6653_fold__graph_Osimps,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
      ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A1),A22))
    <=> ( ( ( A1 = bot_bot(set(A)) )
          & ( A22 = Z2 ) )
        | ? [X4: A,A13: set(A),Y5: B] :
            ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),A13) )
            & ( A22 = aa(B,B,aa(A,fun(B,B),F3,X4),Y5) )
            & ~ pp(aa(set(A),bool,member(A,X4),A13))
            & pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A13),Y5)) ) ) ) ).

% fold_graph.simps
tff(fact_6654_fold__graph_OemptyI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z2: B] : pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,bot_bot(set(A))),Z2)) ).

% fold_graph.emptyI
tff(fact_6655_empty__fold__graphE,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z2: B,X: B] :
      ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,bot_bot(set(A))),X))
     => ( X = Z2 ) ) ).

% empty_fold_graphE
tff(fact_6656_sum__mset_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,fun(C,A)),B4: multiset(C),A5: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_akc(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),G3),B4)),A5)) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_akd(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),G3),A5)),B4)) ) ).

% sum_mset.swap
tff(fact_6657_comp__fun__commute__on_Ofold__graph__determ,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z2: B,X: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A5),X))
         => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A5),Y))
           => ( Y = X ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_determ
tff(fact_6658_sum__mset_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),H: fun(B,A),A5: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_du(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),A5))),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,H),A5))) ) ).

% sum_mset.distrib
tff(fact_6659_sum__mset__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F3: fun(B,A),M6: multiset(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F3),M6))) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dy(A,fun(fun(B,A),fun(B,A)),C2),F3)),M6)) ) ).

% sum_mset_distrib_left
tff(fact_6660_sum__mset__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),M6: multiset(B),C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F3),M6))),C2) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_dx(fun(B,A),fun(A,fun(B,A)),F3),C2)),M6)) ) ).

% sum_mset_distrib_right
tff(fact_6661_sum__mset__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & times(A)
        & semiring_0(B) )
     => ! [F3: fun(A,B),A5: multiset(A),G3: fun(C,B),B4: multiset(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),comm_m7189776963980413722m_mset(B,aa(multiset(A),multiset(B),image_mset(A,B,F3),A5))),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,G3),B4))) = comm_m7189776963980413722m_mset(B,aa(multiset(A),multiset(B),image_mset(A,B,aa(multiset(C),fun(A,B),aa(fun(C,B),fun(multiset(C),fun(A,B)),aTP_Lamp_akf(fun(A,B),fun(fun(C,B),fun(multiset(C),fun(A,B))),F3),G3),B4)),A5)) ) ).

% sum_mset_product
tff(fact_6662_size__eq__sum__mset,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),M6) = comm_m7189776963980413722m_mset(nat,aa(multiset(A),multiset(nat),image_mset(A,nat,aTP_Lamp_la(A,nat)),M6)) ).

% size_eq_sum_mset
tff(fact_6663_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A5: set(A),Z2: B,V2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)),V2))
         => ( ~ pp(aa(set(A),bool,member(A,X),A5))
           => ~ ! [Y3: B] :
                  ( ( V2 = aa(B,B,aa(A,fun(B,B),F3,X),Y3) )
                 => ~ pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A5),Y3)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_6664_comp__fun__commute__on_Ofold__equality,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z2: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A5),Y))
         => ( finite_fold(A,B,F3,Z2,A5) = Y ) ) ) ) ).

% comp_fun_commute_on.fold_equality
tff(fact_6665_Finite__Set_Ofold__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,fun(B,B)),Z2: B] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),A5))
       => ( finite_fold(A,B,F3,Z2,A5) = the(B,finite_fold_graph(A,B,F3,Z2,A5)) ) )
      & ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
       => ( finite_fold(A,B,F3,Z2,A5) = Z2 ) ) ) ).

% Finite_Set.fold_def
tff(fact_6666_comp__fun__commute__on_Ofold__graph__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => pp(aa(B,bool,finite_fold_graph(A,B,F3,Z2,A5),finite_fold(A,B,F3,Z2,A5))) ) ) ) ).

% comp_fun_commute_on.fold_graph_fold
tff(fact_6667_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : size_list(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),size_list(A,X,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6668_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F3),Xs),top_top(A)) ) ).

% INF_set_fold
tff(fact_6669_fold__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : fold(B,A,F3,aa(list(B),list(B),filter2(B,P),Xs)) = fold(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_adh(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F3),P),Xs) ).

% fold_filter
tff(fact_6670_foldl__conv__fold,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),S2: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),S2),Xs) = aa(A,A,fold(B,A,aTP_Lamp_adj(fun(A,fun(B,A)),fun(B,fun(A,A)),F3),Xs),S2) ).

% foldl_conv_fold
tff(fact_6671_size__list__estimation,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(A,nat,F3,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),size_list(A,F3,Xs))) ) ) ).

% size_list_estimation
tff(fact_6672_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),G3: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X3)),aa(A,nat,G3,X3))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),size_list(A,F3,Xs)),size_list(A,G3,Xs))) ) ).

% size_list_pointwise
tff(fact_6673_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(A,nat,F3,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),size_list(A,F3,Xs))) ) ) ).

% size_list_estimation'
tff(fact_6674_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),A5) = aa(set(A),set(A),fold(A,set(A),insert(A),Xs),A5) ).

% union_set_fold
tff(fact_6675_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_rm(A,A)),Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_6676_Sup__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,sup_sup(A),Xs),bot_bot(A)) ) ).

% Sup_set_fold
tff(fact_6677_Inf__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,inf_inf(A),Xs),top_top(A)) ) ).

% Inf_set_fold
tff(fact_6678_Inf__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,inf_inf(A),Xs),X) ) ).

% Inf_fin.set_eq_fold
tff(fact_6679_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,sup_sup(A),Xs),X) ) ).

% Sup_fin.set_eq_fold
tff(fact_6680_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S))
       => ( finite_fold(A,B,F3,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,Xs),Y) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_6681_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S))
       => ( finite_fold(A,B,F3,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,aa(list(A),list(A),remdups(A),Xs)),Y) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_6682_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F3),Xs),bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_6683_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat,S2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(A,set(A),set_ord_lessThan(A),S2)))))
           => ( 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)),N2) = infini527867602293511546merate(A,S,N2) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_6684_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I2))))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J))),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_6685_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or5935395276787703475ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_6686_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_6687_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_6688_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_6689_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioo_iff
tff(fact_6690_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_6691_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_6692_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_6693_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_6694_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M: nat,N2: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N2)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ) ).

% enumerate_mono_iff
tff(fact_6695_Int__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C2,D3)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanLessThan
tff(fact_6696_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M: nat,N2: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(set(A),nat,finite_card(A),S)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(set(A),nat,finite_card(A),S)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N2)))
              <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2)) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_6697_le__enumerate,axiom,
    ! [S: set(nat),N2: nat] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),infini527867602293511546merate(nat,S,N2))) ) ).

% le_enumerate
tff(fact_6698_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B2))) ) ) ).

% infinite_Ioo
tff(fact_6699_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_6700_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_6701_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_6702_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_6703_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_6704_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),J))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6705_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,N2)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)))) ) ) ).

% enumerate_step
tff(fact_6706_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,N2: nat,S: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
         => ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N2))) ) ) ) ).

% enumerate_mono
tff(fact_6707_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X6)))
             => ( infini527867602293511546merate(A,X6,I3) = infini527867602293511546merate(A,Y6,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite2(A),X6))
           => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
             => ( ( aa(set(A),nat,finite_card(A),X6) = aa(set(A),nat,finite_card(A),Y6) )
               => ( X6 = Y6 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_6708_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),S2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(set(A),bool,member(A,S2),S))
           => ? [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(set(A),nat,finite_card(A),S)))
                & ( infini527867602293511546merate(A,S,N5) = S2 ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_6709_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(set(A),nat,finite_card(A),S)))
           => pp(aa(set(A),bool,member(A,infini527867602293511546merate(A,S,N2)),S)) ) ) ) ).

% finite_enumerate_in_set
tff(fact_6710_greaterThanLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or5935395276787703475ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% greaterThanLessThan_def
tff(fact_6711_greaterThanLessThan__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] : set_or5935395276787703475ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)) ) ).

% greaterThanLessThan_eq
tff(fact_6712_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_6713_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_6714_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6715_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_6716_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_6717_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,N2: nat,S: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N2))
         => ( pp(aa(set(A),bool,finite_finite2(A),S))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(set(A),nat,finite_card(A),S)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N2))) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_6718_finite__le__enumerate,axiom,
    ! [S: set(nat),N2: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(set(nat),nat,finite_card(nat),S)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),infini527867602293511546merate(nat,S,N2))) ) ) ).

% finite_le_enumerate
tff(fact_6719_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6720_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6721_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),aa(set(A),nat,finite_card(A),S)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,N2)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)))) ) ) ) ).

% finite_enumerate_step
tff(fact_6722_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)) = 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)),insert(A),infini527867602293511546merate(A,S,zero_zero(nat))),bot_bot(set(A)))),N2) ) ).

% enumerate_Suc'
tff(fact_6723_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X6)))
             => ( infini527867602293511546merate(A,X6,I3) = infini527867602293511546merate(A,Y6,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite2(A),X6))
           => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X6)),aa(set(A),nat,finite_card(A),Y6)))
               => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),Y6)) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_6724_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N2)),aa(set(A),nat,finite_card(A),S)))
           => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_akg(set(A),fun(nat,fun(A,bool)),S),N2)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_6725_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J))),N2) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_6726_Least__eq__0,axiom,
    ! [P: fun(nat,bool)] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_6727_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or5935395276787703475ssThan(code_integer,L,U))) ).

% finite_greaterThanLessThan_integer
tff(fact_6728_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or3652927894154168847AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_6729_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_6730_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_6731_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_6732_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioc_iff
tff(fact_6733_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_6734_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_6735_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_6736_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_6737_Int__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanAtMost
tff(fact_6738_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B2))) ) ) ).

% infinite_Ioc
tff(fact_6739_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),K)) ) ) ).

% Least_le
tff(fact_6740_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
         => ( ! [A6: A] :
                ( pp(aa(A,bool,P,A6))
               => ( ! [B11: A] :
                      ( pp(aa(A,bool,P,B11))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B11)) )
                 => pp(aa(A,bool,Q,A6)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_6741_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A3: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,A3))
         => ( ! [A6: A] :
                ( pp(aa(A,bool,P,A6))
               => ( ! [B11: A] :
                      ( pp(aa(A,bool,P,B11))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B11)) )
                 => pp(aa(A,bool,Q,A6)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder
tff(fact_6742_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( ord_Least(A,P) = X ) ) ) ) ).

% Least_equality
tff(fact_6743_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y4: A] :
                        ( pp(aa(A,bool,P,Y4))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ) ).

% LeastI2_order
tff(fact_6744_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),Z2: A] :
          ( ? [X5: A] :
              ( pp(aa(A,bool,P,X5))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P,Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Ya2)) ) )
                 => ( Y3 = X5 ) ) )
         => ( pp(aa(A,bool,P,Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),Z2)) ) ) ) ).

% Least1_le
tff(fact_6745_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] :
          ( ? [X5: A] :
              ( pp(aa(A,bool,P,X5))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P,Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Ya2)) ) )
                 => ( Y3 = X5 ) ) )
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% Least1I
tff(fact_6746_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( set_or3652927894154168847AtMost(A,A3,B2) = set_or3652927894154168847AtMost(A,C2,D3) )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C2)) )
            | ( ( A3 = C2 )
              & ( B2 = D3 ) ) ) ) ) ).

% Ioc_inj
tff(fact_6747_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_6748_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_6749_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),ord_Least(A,P)))
         => ~ pp(aa(A,bool,P,K)) ) ) ).

% not_less_Least
tff(fact_6750_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% LeastI
tff(fact_6751_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A3: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,A3))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P,X3))
               => pp(aa(A,bool,Q,X3)) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2
tff(fact_6752_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% LeastI_ex
tff(fact_6753_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P,X3))
               => pp(aa(A,bool,Q,X3)) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_ex
tff(fact_6754_Inf__nat__def,axiom,
    ! [X6: set(nat)] : aa(set(nat),nat,complete_Inf_Inf(nat),X6) = ord_Least(nat,aTP_Lamp_akh(set(nat),fun(nat,bool),X6)) ).

% Inf_nat_def
tff(fact_6755_Least__Suc2,axiom,
    ! [P: fun(nat,bool),N2: nat,Q: fun(nat,bool),M: nat] :
      ( pp(aa(nat,bool,P,N2))
     => ( pp(aa(nat,bool,Q,M))
       => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
         => ( ! [K2: nat] :
                ( pp(aa(nat,bool,P,aa(nat,nat,suc,K2)))
              <=> pp(aa(nat,bool,Q,K2)) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_6756_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(6)
tff(fact_6757_Least__Suc,axiom,
    ! [P: fun(nat,bool),N2: nat] :
      ( pp(aa(nat,bool,P,N2))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_aki(fun(nat,bool),fun(nat,bool),P))) ) ) ) ).

% Least_Suc
tff(fact_6758_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = bot_bot(set(A)) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),A3)) ) ) ) ).

% Ioc_disjoint
tff(fact_6759_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_6760_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(8)
tff(fact_6761_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_6762_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_6763_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
         => ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
           => ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),P)) ) ) ) ) ).

% Least_Min
tff(fact_6764_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(2)
tff(fact_6765_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),P: fun(A,bool)] : bleast(A,S,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_akj(set(A),fun(fun(A,bool),fun(A,bool)),S),P)) ) ).

% Bleast_def
tff(fact_6766_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),P: fun(A,bool)] : abort_Bleast(A,S,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_akj(set(A),fun(fun(A,bool),fun(A,bool)),S),P)) ) ).

% abort_Bleast_def
tff(fact_6767_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] : infini527867602293511546merate(A,S,zero_zero(nat)) = ord_Least(A,aTP_Lamp_akk(set(A),fun(A,bool),S)) ) ).

% enumerate_0
tff(fact_6768_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_6769_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_6770_greaterThanAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or3652927894154168847AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% greaterThanAtMost_def
tff(fact_6771_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or3652927894154168847AtMost(nat,M,N2))) ) ) ) ).

% sum.head
tff(fact_6772_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_6773_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or3652927894154168847AtMost(nat,M,N2))) ) ) ) ).

% prod.head
tff(fact_6774_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_6775_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_6776_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] : set_or3652927894154168847AtMost(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_6777_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_6778_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_6779_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_6780_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_6781_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),S: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( ? [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),S))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,member(A,Xa4),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa4)) ) )
           => ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_akl(fun(A,B),fun(set(A),fun(B,bool)),F3),S)) = aa(A,B,F3,ord_Least(A,aTP_Lamp_akm(set(A),fun(A,bool),S))) ) ) ) ) ).

% Least_mono
tff(fact_6782_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,bool)] : ord_Least(A,P) = the(A,aTP_Lamp_akn(fun(A,bool),fun(A,bool),P)) ) ).

% Least_def
tff(fact_6783_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I2)),J))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6784_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_6785_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_6786_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_6787_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
         => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_akg(set(A),fun(nat,fun(A,bool)),S),N2)) ) ) ) ).

% enumerate_Suc''
tff(fact_6788_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N2: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)) = 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)),insert(A),ord_Least(A,aTP_Lamp_akk(set(A),fun(A,bool),S))),bot_bot(set(A)))),N2) ) ).

% enumerate_Suc
tff(fact_6789_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A6: A,B5: A,X3: A] :
              ( pp(aa(set(A),bool,member(A,A6),S))
             => ( pp(aa(set(A),bool,member(A,B5),S))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B5))
                   => pp(aa(set(A),bool,member(A,X3),S)) ) ) ) )
         => ? [A6: A,B5: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = aa(A,set(A),set_ord_lessThan(A),B5) )
              | ( S = aa(A,set(A),set_ord_atMost(A),B5) )
              | ( S = aa(A,set(A),set_ord_greaterThan(A),A6) )
              | ( S = aa(A,set(A),set_ord_atLeast(A),A6) )
              | ( S = set_or5935395276787703475ssThan(A,A6,B5) )
              | ( S = set_or3652927894154168847AtMost(A,A6,B5) )
              | ( S = set_or7035219750837199246ssThan(A,A6,B5) )
              | ( S = set_or1337092689740270186AtMost(A,A6,B5) ) ) ) ) ).

% interval_cases
tff(fact_6790_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R2),A3),B2))
    <=> ? [Z6: list(product_prod(A,B))] :
          ( pp(aa(set(list(product_prod(A,B))),bool,member(list(product_prod(A,B)),Z6),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ako(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R2))))
          & ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z6) = A3 )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z6) = B2 ) ) ) ).

% list.in_rel
tff(fact_6791_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or3652927894154168847AtMost(code_integer,L,U))) ).

% finite_greaterThanAtMost_integer
tff(fact_6792_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_atLeast(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),I2)) ) ) ).

% atLeast_iff
tff(fact_6793_atLeast__empty__triv,axiom,
    ! [A: $tType] : aa(set(A),set(set(A)),set_ord_atLeast(set(A)),bot_bot(set(A))) = top_top(set(set(A))) ).

% atLeast_empty_triv
tff(fact_6794_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),X)),aa(A,set(A),set_ord_atLeast(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% atLeast_subset_iff
tff(fact_6795_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atLeast(A),L3)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L3),L)) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_6796_Int__atLeastAtMostR2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),A3)),set_or1337092689740270186AtMost(A,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),D3) ) ).

% Int_atLeastAtMostR2
tff(fact_6797_Int__atLeastAtMostL2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(A,set(A),set_ord_atLeast(A),C2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),B2) ) ).

% Int_atLeastAtMostL2
tff(fact_6798_list_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,list_all2(A,B,R2),fequal(bool)),aTP_Lamp_akp(list(A),bool)),aTP_Lamp_akq(list(B),bool))) ).

% list.disc_transfer(1)
tff(fact_6799_list_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,list_all2(A,B,R2),fequal(bool)),aTP_Lamp_si(list(A),bool)),aTP_Lamp_ads(list(B),bool))) ).

% list.disc_transfer(2)
tff(fact_6800_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),X) = top_top(set(A)) )
        <=> ( X = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_6801_not__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atLeast(A),L) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_6802_Misc_Olist__all2__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),L: list(A),L3: list(B),Q: fun(list(A),fun(list(B),bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),L),L3))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,nil(A)),nil(B)))
       => ( ! [X3: A,X7: B,Ls: list(A),Ls3: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),X7))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Ls),Ls3))
               => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,Ls),Ls3))
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ls)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X7),Ls3))) ) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,L),L3)) ) ) ) ).

% Misc.list_all2_induct
tff(fact_6803_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As: list(A),F3: fun(C,B),Bs: list(C)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),As),aa(list(C),list(B),map(C,B,F3),Bs)))
    <=> pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),list_all2(A,C,aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_akr(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F3)),As),Bs)) ) ).

% list_all2_map2
tff(fact_6804_list__all2__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F3: fun(C,A),As: list(C),Bs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(C),list(A),map(C,A,F3),As)),Bs))
    <=> pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),list_all2(C,B,aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_aks(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F3)),As),Bs)) ) ).

% list_all2_map1
tff(fact_6805_list_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: list(A),Y: list(B)] :
      ( pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),list_all2(C,B,Sb),aa(list(A),list(C),map(A,C,I2),X)),Y))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% list.rel_map(1)
tff(fact_6806_list_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: list(A),G3: fun(B,C),Y: list(B)] :
      ( pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),list_all2(A,C,Sa),X),aa(list(B),list(C),map(B,C,G3),Y)))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahi(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G3)),X),Y)) ) ).

% list.rel_map(2)
tff(fact_6807_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),H3))) ) ).

% not_Ici_le_Iic
tff(fact_6808_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),aa(A,set(A),set_ord_atLeast(A),L3))) ) ).

% not_Iic_le_Ici
tff(fact_6809_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L3: A,H3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),set_or1337092689740270186AtMost(A,L3,H3))) ) ).

% not_Ici_le_Icc
tff(fact_6810_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra2))
     => pp(aa(fun(list(A),fun(list(B),bool)),bool,aa(fun(list(A),fun(list(B),bool)),fun(fun(list(A),fun(list(B),bool)),bool),ord_less_eq(fun(list(A),fun(list(B),bool))),list_all2(A,B,R2)),list_all2(A,B,Ra2))) ) ).

% list.rel_mono
tff(fact_6811_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_6812_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atLeast(A),L))) ) ).

% not_UNIV_le_Ici
tff(fact_6813_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_atLeast(A),A3))) ) ).

% Ioi_le_Ico
tff(fact_6814_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P3),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3))) ) ) ).

% list_all2_nthD
tff(fact_6815_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P3),aa(list(B),nat,size_size(list(B)),Ys2)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3))) ) ) ).

% list_all2_nthD2
tff(fact_6816_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),A3)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A3),N5)),aa(nat,B,nth(B,B2),N5))) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),A3),B2)) ) ) ).

% list_all2_all_nthI
tff(fact_6817_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,B,nth(B,Ys2),I4))) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_6818_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_6819_ivl__disj__int__one_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(8)
tff(fact_6820_atLeastAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or1337092689740270186AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% atLeastAtMost_def
tff(fact_6821_atLeastLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or7035219750837199246ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% atLeastLessThan_def
tff(fact_6822_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),A3)),aa(A,set(A),set_ord_greaterThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6823_ivl__disj__int__one_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(6)
tff(fact_6824_product__lists__set,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_aku(list(list(A)),fun(list(A),bool),Xss)) ).

% product_lists_set
tff(fact_6825_atMost__Int__atLeast,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N2: 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),N2)),aa(A,set(A),set_ord_atLeast(A),N2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),N2),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_6826_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)),insert(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_6827_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_6828_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_6829_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_6830_atLeast__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_6831_UN__atLeast__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atLeast_UNIV
tff(fact_6832_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P),X4)) ) ) ) ).

% list_all2_iff
tff(fact_6833_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A5: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A5,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A5,bNF_rel_fun(A,B,A,B,A5,A5)),plus_plus(A)),plus_plus(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A5,bNF_rel_fun(A,B,A,B,A5,A5)),times_times(A)),times_times(B)))
             => pp(aa(fun(fun(D,B),fun(B,fun(list(D),B))),bool,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B4,A5),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A5,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B4),A5))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).

% horner_sum_transfer
tff(fact_6834_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_akv(A,filter(A))),top_top(set(A)))) ) ) ).

% at_top_def
tff(fact_6835_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [A5: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A5,one_one(A)),one_one(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A5,bNF_rel_fun(A,B,A,B,A5,A5)),times_times(A)),times_times(B)))
           => pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A5),A5),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B))) ) ) ) ).

% prod_list_transfer
tff(fact_6836_prod__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ).

% prod_list.Cons
tff(fact_6837_prod__list_Oappend,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A),Ys2: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xs)),aa(list(A),A,groups5270119922927024881d_list(A),Ys2)) ) ).

% prod_list.append
tff(fact_6838_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_6839_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = aa(A,A,foldr(A,A,times_times(A),Xs),one_one(A)) ) ).

% prod_list.eq_foldr
tff(fact_6840_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_akw(A,filter(A))),aa(A,set(A),set_ord_atLeast(A),C2))) ) ).

% at_top_sub
tff(fact_6841_finite__refines__card__le,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A5,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
       => ( equiv_equiv(A,A5,R2)
         => ( equiv_equiv(A,A5,S)
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A5,S))),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A5,R2)))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_6842_congruent__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,B)] :
      ( equiv_congruent(A,B,R3,F3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_aew(fun(A,B),fun(A,fun(A,bool)),F3)),X4)) ) ) ).

% congruent_def
tff(fact_6843_in__quotient__imp__subset,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A5)) ) ) ).

% in_quotient_imp_subset
tff(fact_6844_equiv__type,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( equiv_equiv(A,A5,R3)
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% equiv_type
tff(fact_6845_quotient__eqI,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A),Y6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y6),equiv_quotient(A,A5,R3)))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( pp(aa(set(A),bool,member(A,Y),Y6))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_6846_quotient__eq__iff,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A),Y6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y6),equiv_quotient(A,A5,R3)))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( pp(aa(set(A),bool,member(A,Y),Y6))
             => ( ( X6 = Y6 )
              <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_6847_in__quotient__imp__closed,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
       => ( pp(aa(set(A),bool,member(A,X),X6))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
           => pp(aa(set(A),bool,member(A,Y),X6)) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_6848_congruentI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,B)] :
      ( ! [Y3: A,Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
         => ( aa(A,B,F3,Y3) = aa(A,B,F3,Z3) ) )
     => equiv_congruent(A,B,R3,F3) ) ).

% congruentI
tff(fact_6849_congruentD,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,B),Y: A,Z2: A] :
      ( equiv_congruent(A,B,R3,F3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R3))
       => ( aa(A,B,F3,Y) = aa(A,B,F3,Z2) ) ) ) ).

% congruentD
tff(fact_6850_UN__equiv__class__type,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R3: set(product_prod(A,A)),F3: fun(A,set(B)),X6: set(A),B4: set(set(B))] :
      ( equiv_equiv(A,A5,R3)
     => ( equiv_congruent(A,set(B),R3,F3)
       => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => pp(aa(set(set(B)),bool,member(set(B),aa(A,set(B),F3,X3)),B4)) )
           => pp(aa(set(set(B)),bool,member(set(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X6))),B4)) ) ) ) ) ).

% UN_equiv_class_type
tff(fact_6851_in__quotient__imp__non__empty,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
       => ( X6 != bot_bot(set(A)) ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_6852_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(A,A)),F3: fun(A,set(B)),X6: set(A),Y6: set(A)] :
      ( equiv_equiv(A,A5,R3)
     => ( equiv_congruent(A,set(B),R3,F3)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X6)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),Y6)) )
         => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
           => ( pp(aa(set(set(A)),bool,member(set(A),Y6),equiv_quotient(A,A5,R3)))
             => ( ! [X3: A,Y3: A] :
                    ( pp(aa(set(A),bool,member(A,X3),A5))
                   => ( pp(aa(set(A),bool,member(A,Y3),A5))
                     => ( ( aa(A,set(B),F3,X3) = aa(A,set(B),F3,Y3) )
                       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3)) ) ) )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_6853_equiv__class__self,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),A3: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))) ) ) ).

% equiv_class_self
tff(fact_6854_quotient__disj,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A),Y6: set(A)] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y6),equiv_quotient(A,A5,R3)))
         => ( ( X6 = Y6 )
            | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X6),Y6) = bot_bot(set(A)) ) ) ) ) ) ).

% quotient_disj
tff(fact_6855_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(A,A)),F3: fun(A,set(B)),A3: A] :
      ( equiv_equiv(A,A5,R3)
     => ( equiv_congruent(A,set(B),R3,F3)
       => ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))) = aa(A,set(B),F3,A3) ) ) ) ) ).

% UN_equiv_class
tff(fact_6856_finite__refines__finite,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A5,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
       => ( equiv_equiv(A,A5,R2)
         => ( equiv_equiv(A,A5,S)
           => pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A5,S))) ) ) ) ) ).

% finite_refines_finite
tff(fact_6857_eq__equiv__class,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,A5: set(A)] :
      ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A5,R3)
       => ( pp(aa(set(A),bool,member(A,B2),A5))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ).

% eq_equiv_class
tff(fact_6858_equiv__class__eq,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_6859_eq__equiv__class__iff,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( pp(aa(set(A),bool,member(A,Y),A5))
         => ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_6860_equiv__class__eq__iff,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
      <=> ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) )
          & pp(aa(set(A),bool,member(A,X),A5))
          & pp(aa(set(A),bool,member(A,Y),A5)) ) ) ) ).

% equiv_class_eq_iff
tff(fact_6861_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( pp(aa(set(A),bool,member(A,Y),A5))
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),R3) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))),R3) )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_6862_refines__equiv__class__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A)),A5: set(A),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( equiv_equiv(A,A5,R2)
       => ( equiv_equiv(A,A5,S)
         => ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_6863_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A)),A5: set(A),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( equiv_equiv(A,A5,R2)
       => ( equiv_equiv(A,A5,S)
         => ( aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_6864_refines__equiv__image__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A)),A5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( equiv_equiv(A,A5,R2)
       => ( equiv_equiv(A,A5,S)
         => ( aa(set(set(A)),set(set(A)),image2(set(A),set(A),image(A,A,S)),equiv_quotient(A,A5,R2)) = equiv_quotient(A,A5,S) ) ) ) ) ).

% refines_equiv_image_eq
tff(fact_6865_equiv__class__subset,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))) ) ) ).

% equiv_class_subset
tff(fact_6866_subset__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),B2: A,A3: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))))
       => ( pp(aa(set(A),bool,member(A,B2),A5))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ).

% subset_equiv_class
tff(fact_6867_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A,A3: A,B2: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ).

% equiv_class_nondisjoint
tff(fact_6868_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R3)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),X6))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_6869_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType,A16: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),A3: B] :
      ( equiv_equiv(A,A16,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F3)
         => ( pp(aa(set(B),bool,member(B,A3),A24))
           => equiv_congruent(A,set(C),R12,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_akx(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F3),A3)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_6870_UN__equiv__class2,axiom,
    ! [A: $tType,C: $tType,B: $tType,A16: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),A1: A,A22: B] :
      ( equiv_equiv(A,A16,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F3)
         => ( pp(aa(set(A),bool,member(A,A1),A16))
           => ( pp(aa(set(B),bool,member(B,A22),A24))
             => ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_akx(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F3),A22)),aa(set(A),set(A),image(A,A,R12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F3,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_6871_congruent2D,axiom,
    ! [A: $tType,C: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C)),Y1: A,Z1: A,Y2: B,Z22: B] :
      ( equiv_congruent2(A,B,C,R12,R23,F3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1)),R12))
       => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z22)),R23))
         => ( aa(B,C,aa(A,fun(B,C),F3,Y1),Y2) = aa(B,C,aa(A,fun(B,C),F3,Z1),Z22) ) ) ) ) ).

% congruent2D
tff(fact_6872_congruent2I_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( ! [Y12: A,Z12: A,Y22: B,Z23: B] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y12),Z12)),R12))
         => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z23)),R23))
           => ( aa(B,C,aa(A,fun(B,C),F3,Y12),Y22) = aa(B,C,aa(A,fun(B,C),F3,Z12),Z23) ) ) )
     => equiv_congruent2(A,B,C,R12,R23,F3) ) ).

% congruent2I'
tff(fact_6873_congruent2I,axiom,
    ! [C: $tType,B: $tType,A: $tType,A16: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( equiv_equiv(A,A16,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( ! [Y3: A,Z3: A,W: B] :
              ( pp(aa(set(B),bool,member(B,W),A24))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R12))
               => ( aa(B,C,aa(A,fun(B,C),F3,Y3),W) = aa(B,C,aa(A,fun(B,C),F3,Z3),W) ) ) )
         => ( ! [Y3: B,Z3: B,W: A] :
                ( pp(aa(set(A),bool,member(A,W),A16))
               => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y3),Z3)),R23))
                 => ( aa(B,C,aa(A,fun(B,C),F3,W),Y3) = aa(B,C,aa(A,fun(B,C),F3,W),Z3) ) ) )
           => equiv_congruent2(A,B,C,R12,R23,F3) ) ) ) ) ).

% congruent2I
tff(fact_6874_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(A,A)),F3: fun(A,fun(A,B))] :
      ( equiv_equiv(A,A5,R3)
     => ( ! [Y3: A,Z3: A] :
            ( pp(aa(set(A),bool,member(A,Y3),A5))
           => ( pp(aa(set(A),bool,member(A,Z3),A5))
             => ( aa(A,B,aa(A,fun(A,B),F3,Y3),Z3) = aa(A,B,aa(A,fun(A,B),F3,Z3),Y3) ) ) )
       => ( ! [Y3: A,Z3: A,W: A] :
              ( pp(aa(set(A),bool,member(A,W),A5))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
               => ( aa(A,B,aa(A,fun(A,B),F3,W),Y3) = aa(A,B,aa(A,fun(A,B),F3,W),Z3) ) ) )
         => equiv_congruent2(A,A,B,R3,R3,F3) ) ) ) ).

% congruent2_commuteI
tff(fact_6875_congruent2__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( equiv_congruent2(A,B,C,R12,R23,F3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R12))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,fun(B,C)),fun(A,fun(A,bool)),aTP_Lamp_akz(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),R23),F3)),X4)) ) ) ).

% congruent2_def
tff(fact_6876_UN__equiv__class__type2,axiom,
    ! [A: $tType,B: $tType,C: $tType,A16: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),X13: set(A),X23: set(B),B4: set(set(C))] :
      ( equiv_equiv(A,A16,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F3)
         => ( pp(aa(set(set(A)),bool,member(set(A),X13),equiv_quotient(A,A16,R12)))
           => ( pp(aa(set(set(B)),bool,member(set(B),X23),equiv_quotient(B,A24,R23)))
             => ( ! [X12: A,X22: B] :
                    ( pp(aa(set(A),bool,member(A,X12),A16))
                   => ( pp(aa(set(B),bool,member(B,X22),A24))
                     => pp(aa(set(set(C)),bool,member(set(C),aa(B,set(C),aa(A,fun(B,set(C)),F3,X12),X22)),B4)) ) )
               => pp(aa(set(set(C)),bool,member(set(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(set(B),fun(A,set(C)),aTP_Lamp_ala(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),F3),X23)),X13))),B4)) ) ) ) ) ) ) ).

% UN_equiv_class_type2
tff(fact_6877_proj__iff,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),A5))
       => ( ( aa(A,set(A),equiv_proj(A,A,R3),X) = aa(A,set(A),equiv_proj(A,A,R3),Y) )
        <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ).

% proj_iff
tff(fact_6878_relImage__proj,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,A5,R2)
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),bNF_Gr4221423524335903396lImage(A,set(A),R2,equiv_proj(A,A,R2))),id_on(set(A),equiv_quotient(A,A5,R2)))) ) ).

% relImage_proj
tff(fact_6879_proj__def,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),X: B] : aa(B,set(A),equiv_proj(B,A,R3),X) = aa(set(B),set(A),image(B,A,R3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))) ).

% proj_def
tff(fact_6880_remove__def,axiom,
    ! [A: $tType,X: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% remove_def
tff(fact_6881_lenlex__append2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys2: list(A)] :
      ( irrefl(A,R2)
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys2))),lenlex(A,R2)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys2)),lenlex(A,R2))) ) ) ).

% lenlex_append2
tff(fact_6882_member__remove,axiom,
    ! [A: $tType,X: A,Y: A,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Y),A5)))
    <=> ( pp(aa(set(A),bool,member(A,X),A5))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_6883_lexord__same__pref__if__irrefl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( irrefl(A,R3)
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R3)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys2),Zs)),lexord(A,R3))) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_6884_irreflI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [A6: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),A6)),R2))
     => irrefl(A,R2) ) ).

% irreflI
tff(fact_6885_irrefl__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( irrefl(A,R3)
    <=> ! [A8: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A8),A8)),R3)) ) ).

% irrefl_def
tff(fact_6886_lexl__not__refl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: list(A)] :
      ( irrefl(A,R3)
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),X)),lex(A,R3))) ) ).

% lexl_not_refl
tff(fact_6887_irrefl__diff__Id,axiom,
    ! [A: $tType,R3: 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))),R3),id2(A))) ).

% irrefl_diff_Id
tff(fact_6888_irrefl__distinct,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( irrefl(A,R3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_sn(A,fun(A,bool))),X4)) ) ) ).

% irrefl_distinct
tff(fact_6889_inter__coset__fold,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),coset(A,Xs)) = aa(set(A),set(A),fold(A,set(A),remove(A),Xs),A5) ).

% inter_coset_fold
tff(fact_6890_relImage__Gr,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),A5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))
     => ( bNF_Gr4221423524335903396lImage(A,B,R2,F3) = relcomp(B,A,B,converse(A,B,bNF_Gr(A,B,A5,F3)),relcomp(A,A,B,R2,bNF_Gr(A,B,A5,F3))) ) ) ).

% relImage_Gr
tff(fact_6891_converse__converse,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : converse(B,A,converse(A,B,R3)) = R3 ).

% converse_converse
tff(fact_6892_converse__inject,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),S2: set(product_prod(B,A))] :
      ( ( converse(B,A,R3) = converse(B,A,S2) )
    <=> ( R3 = S2 ) ) ).

% converse_inject
tff(fact_6893_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R3: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R3)))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R3)) ) ).

% converse_iff
tff(fact_6894_Field__converse,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),converse(A,A,R3)) = aa(set(product_prod(A,A)),set(A),field2(A),R3) ).

% Field_converse
tff(fact_6895_converse__mono,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),S2: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),converse(B,A,R3)),converse(B,A,S2)))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),R3),S2)) ) ).

% converse_mono
tff(fact_6896_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_6897_converse__Id,axiom,
    ! [A: $tType] : converse(A,A,id2(A)) = id2(A) ).

% converse_Id
tff(fact_6898_refl__on__converse,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A5,converse(A,A,R3))
    <=> refl_on(A,A5,R3) ) ).

% refl_on_converse
tff(fact_6899_finite__converse,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),converse(B,A,R3)))
    <=> pp(aa(set(product_prod(B,A)),bool,finite_finite2(product_prod(B,A)),R3)) ) ).

% finite_converse
tff(fact_6900_total__on__converse,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( total_on(A,A5,converse(A,A,R3))
    <=> total_on(A,A5,R3) ) ).

% total_on_converse
tff(fact_6901_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_6902_antisym__converse,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( antisym(A,converse(A,A,R3))
    <=> antisym(A,R3) ) ).

% antisym_converse
tff(fact_6903_converse__Id__on,axiom,
    ! [A: $tType,A5: set(A)] : converse(A,A,id_on(A,A5)) = id_on(A,A5) ).

% converse_Id_on
tff(fact_6904_converse__inv__image,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,B)),F3: fun(A,B)] : converse(A,A,inv_image(B,A,R2,F3)) = inv_image(B,A,converse(B,B,R2),F3) ).

% converse_inv_image
tff(fact_6905_card__inverse,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),converse(B,A,R2)) = aa(set(product_prod(B,A)),nat,finite_card(product_prod(B,A)),R2) ).

% card_inverse
tff(fact_6906_pair__set__inverse,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,bool))] : converse(B,A,aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P))) ).

% pair_set_inverse
tff(fact_6907_below__Id__inv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),converse(A,A,R2)),id2(A)))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),id2(A))) ) ).

% below_Id_inv
tff(fact_6908_converse__subset__swap,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),converse(B,A,S2)))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),converse(A,B,R3)),S2)) ) ).

% converse_subset_swap
tff(fact_6909_converse__unfold,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A))] : converse(B,A,R3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_alb(set(product_prod(B,A)),fun(A,fun(B,bool)),R3))) ).

% converse_unfold
tff(fact_6910_trancl__converseI,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R3))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R3)))) ) ).

% trancl_converseI
tff(fact_6911_trancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R3))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R3)))) ) ).

% trancl_converseD
tff(fact_6912_in__listrel1__converse,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),listrel1(A,converse(A,A,R3))))
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),converse(list(A),list(A),listrel1(A,R3)))) ) ).

% in_listrel1_converse
tff(fact_6913_converse_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: B,A22: A,R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22)),converse(A,B,R3)))
     => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A22),A1)),R3)) ) ).

% converse.cases
tff(fact_6914_converse_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: B,A22: A,R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22)),converse(A,B,R3)))
    <=> ? [A8: A,B13: B] :
          ( ( A1 = B13 )
          & ( A22 = A8 )
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B13)),R3)) ) ) ).

% converse.simps
tff(fact_6915_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R3: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R3)))
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R3)) ) ).

% converseD
tff(fact_6916_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R3: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Yx),converse(B,A,R3)))
     => ~ ! [X3: B,Y3: A] :
            ( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),X3) )
           => ~ pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y3)),R3)) ) ) ).

% converseE
tff(fact_6917_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),converse(A,B,R3))) ) ).

% converseI
tff(fact_6918_rtrancl__converseI,axiom,
    ! [A: $tType,Y: A,X: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R3)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R3)))) ) ).

% rtrancl_converseI
tff(fact_6919_rtrancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R3))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R3))) ) ).

% rtrancl_converseD
tff(fact_6920_finite__wf__eq__wf__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( wf(A,converse(A,A,R2))
      <=> wf(A,R2) ) ) ).

% finite_wf_eq_wf_converse
tff(fact_6921_converse__Times,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B4: set(A)] : converse(B,A,product_Sigma(B,A,A5,aTP_Lamp_mi(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),A5)) ).

% converse_Times
tff(fact_6922_converse__Int,axiom,
    ! [A: $tType,B: $tType,R3: 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))),R3),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,R3)),converse(B,A,S2)) ).

% converse_Int
tff(fact_6923_converse__relcomp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(B,C)),S2: set(product_prod(C,A))] : converse(B,A,relcomp(B,C,A,R3,S2)) = relcomp(A,C,B,converse(C,A,S2),converse(B,C,R3)) ).

% converse_relcomp
tff(fact_6924_converse__Un,axiom,
    ! [A: $tType,B: $tType,R3: 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))),R3),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,R3)),converse(B,A,S2)) ).

% converse_Un
tff(fact_6925_converse__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R3),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_alc(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R3)),S)) ).

% converse_UNION
tff(fact_6926_converse__INTER,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Inf_Inf(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R3),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_alc(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R3)),S)) ).

% converse_INTER
tff(fact_6927_Image__subset__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(B,A)),A5: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R3),A5)),B4))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,converse(B,A,R3)),aa(set(A),set(A),uminus_uminus(set(A)),B4))))) ) ).

% Image_subset_eq
tff(fact_6928_refl__on__comp__subset,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A5,R3)
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),relcomp(A,A,A,converse(A,A,R3),R3))) ) ).

% refl_on_comp_subset
tff(fact_6929_subset__code_I2_J,axiom,
    ! [B: $tType,A5: set(B),Ys2: list(B)] :
      ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),coset(B,Ys2)))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),aa(list(B),set(B),set2(B),Ys2)))
         => ~ pp(aa(set(B),bool,member(B,X4),A5)) ) ) ).

% subset_code(2)
tff(fact_6930_union__coset__filter,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A5) = coset(A,aa(list(A),list(A),filter2(A,aTP_Lamp_cj(set(A),fun(A,bool),A5)),Xs)) ).

% union_coset_filter
tff(fact_6931_irrefl__tranclI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A] :
      ( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R3)),transitive_rtrancl(A,R3)) = bot_bot(set(product_prod(A,A))) )
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),transitive_trancl(A,R3))) ) ).

% irrefl_tranclI
tff(fact_6932_subset__code_I3_J,axiom,
    ! [C: $tType] : ~ pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),coset(C,nil(C))),aa(list(C),set(C),set2(C),nil(C)))) ).

% subset_code(3)
tff(fact_6933_minus__coset__filter,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),coset(A,Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5)),Xs)) ).

% minus_coset_filter
tff(fact_6934_relInvImage__Gr,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),B4: set(A),A5: set(B),F3: fun(B,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))))
     => ( bNF_Gr7122648621184425601vImage(B,A,A5,R2,F3) = relcomp(B,A,B,bNF_Gr(B,A,A5,F3),relcomp(A,A,B,R2,converse(B,A,bNF_Gr(B,A,A5,F3)))) ) ) ).

% relInvImage_Gr
tff(fact_6935_trans__wf__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
     => ( wf(A,R3)
      <=> ! [A8: 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))),R3),product_Sigma(A,A,aa(set(A),set(A),image(A,A,converse(A,A,R3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A8),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_ald(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),A8)))) ) ) ).

% trans_wf_iff
tff(fact_6936_Image__INT__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(B,A)),A5: set(C),B4: fun(C,set(B))] :
      ( single_valued(A,B,converse(B,A,R3))
     => ( ( A5 != bot_bot(set(C)) )
       => ( aa(set(B),set(A),image(B,A,R3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ahc(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R3),B4)),A5)) ) ) ) ).

% Image_INT_eq
tff(fact_6937_trans__converse,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,converse(A,A,R3))
    <=> trans(A,R3) ) ).

% trans_converse
tff(fact_6938_single__valued__Id__on,axiom,
    ! [A: $tType,A5: set(A)] : single_valued(A,A,id_on(A,A5)) ).

% single_valued_Id_on
tff(fact_6939_trans__Id__on,axiom,
    ! [A: $tType,A5: set(A)] : trans(A,id_on(A,A5)) ).

% trans_Id_on
tff(fact_6940_trans__inv__image,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),F3: fun(B,A)] :
      ( trans(A,R3)
     => trans(B,inv_image(A,B,R3,F3)) ) ).

% trans_inv_image
tff(fact_6941_single__valued__relcomp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( single_valued(A,B,R3)
     => ( single_valued(B,C,S2)
       => single_valued(A,C,relcomp(A,B,C,R3,S2)) ) ) ).

% single_valued_relcomp
tff(fact_6942_single__valued__empty,axiom,
    ! [B: $tType,A: $tType] : single_valued(A,B,bot_bot(set(product_prod(A,B)))) ).

% single_valued_empty
tff(fact_6943_trans__empty,axiom,
    ! [A: $tType] : trans(A,bot_bot(set(product_prod(A,A)))) ).

% trans_empty
tff(fact_6944_single__valued__Id,axiom,
    ! [A: $tType] : single_valued(A,A,id2(A)) ).

% single_valued_Id
tff(fact_6945_trans__Id,axiom,
    ! [A: $tType] : trans(A,id2(A)) ).

% trans_Id
tff(fact_6946_trans__reflclI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
     => trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),id2(A))) ) ).

% trans_reflclI
tff(fact_6947_single__valued__inter1,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( single_valued(A,B,R2)
     => 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))),R2),S)) ) ).

% single_valued_inter1
tff(fact_6948_single__valued__inter2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( single_valued(A,B,R2)
     => 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),R2)) ) ).

% single_valued_inter2
tff(fact_6949_trans__Int,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( trans(A,R3)
     => ( 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))),R3),S2)) ) ) ).

% trans_Int
tff(fact_6950_trans__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( trans(A,R3)
     => 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))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% trans_Restr
tff(fact_6951_lenlex__trans,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A)),Z2: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lenlex(A,R3)))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2)),lenlex(A,R3)))
       => ( trans(A,R3)
         => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Z2)),lenlex(A,R3))) ) ) ) ).

% lenlex_trans
tff(fact_6952_single__valued__def,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B))] :
      ( single_valued(A,B,R3)
    <=> ! [X4: A,Y5: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y5)),R3))
         => ! [Z6: B] :
              ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z6)),R3))
             => ( Y5 = Z6 ) ) ) ) ).

% single_valued_def
tff(fact_6953_single__valuedI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] :
      ( ! [X3: A,Y3: B,Z3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R3))
         => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3)),R3))
           => ( Y3 = Z3 ) ) )
     => single_valued(A,B,R3) ) ).

% single_valuedI
tff(fact_6954_single__valuedD,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B)),X: A,Y: B,Z2: B] :
      ( single_valued(A,B,R3)
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R3))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2)),R3))
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
tff(fact_6955_transD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R3))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R3)) ) ) ) ).

% transD
tff(fact_6956_transE,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R3))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R3)) ) ) ) ).

% transE
tff(fact_6957_transI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A,Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3)),R3)) ) )
     => trans(A,R3) ) ).

% transI
tff(fact_6958_trans__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
    <=> ! [X4: A,Y5: A,Z6: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z6)),R3))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z6)),R3)) ) ) ) ).

% trans_def
tff(fact_6959_lexord__trans,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A)),Z2: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R3)))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2)),lexord(A,R3)))
       => ( trans(A,R3)
         => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Z2)),lexord(A,R3))) ) ) ) ).

% lexord_trans
tff(fact_6960_trans__INTER,axiom,
    ! [B: $tType,A: $tType,S: set(A),R3: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),S))
         => trans(B,aa(A,set(product_prod(B,B)),R3,X3)) )
     => 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)),R3),S))) ) ).

% trans_INTER
tff(fact_6961_trans__O__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,R3)),R3)) ) ).

% trans_O_subset
tff(fact_6962_single__valued__subset,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S2))
     => ( single_valued(A,B,S2)
       => single_valued(A,B,R3) ) ) ).

% single_valued_subset
tff(fact_6963_single__valued__confluent,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( single_valued(A,A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_rtrancl(A,R3)))
            | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y)),transitive_rtrancl(A,R3))) ) ) ) ) ).

% single_valued_confluent
tff(fact_6964_single__valued__below__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),id2(A)))
     => single_valued(A,A,R2) ) ).

% single_valued_below_Id
tff(fact_6965_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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),bot_bot(set(product_prod(A,A))))) ).

% trans_singleton
tff(fact_6966_trans__rtrancl__eq__reflcl,axiom,
    ! [A: $tType,A5: set(product_prod(A,A))] :
      ( trans(A,A5)
     => ( transitive_rtrancl(A,A5) = 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))),A5),id2(A)) ) ) ).

% trans_rtrancl_eq_reflcl
tff(fact_6967_trans__diff__Id,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
     => ( antisym(A,R3)
       => 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))),R3),id2(A))) ) ) ).

% trans_diff_Id
tff(fact_6968_trans__join,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_alf(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)),X4)) ) ) ).

% trans_join
tff(fact_6969_bijective__alt,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( bijective(A,B,R2)
    <=> ( single_valued(A,B,R2)
        & single_valued(B,A,converse(A,B,R2)) ) ) ).

% bijective_alt
tff(fact_6970_Image__absorb__rtrancl,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),B4: set(A),C4: set(A)] :
      ( trans(A,A5)
     => ( refl_on(A,B4,A5)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B4))
         => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,A5)),C4) = aa(set(A),set(A),image(A,A,A5),C4) ) ) ) ) ).

% Image_absorb_rtrancl
tff(fact_6971_Image__Int__eq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A5: set(B),B4: set(B)] :
      ( single_valued(A,B,converse(B,A,R2))
     => ( aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,R2),A5)),aa(set(B),set(A),image(B,A,R2),B4)) ) ) ).

% Image_Int_eq
tff(fact_6972_wf__finite__segments,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( irrefl(A,R3)
     => ( trans(A,R3)
       => ( ! [X3: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_alg(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),X3))))
         => wf(A,R3) ) ) ) ).

% wf_finite_segments
tff(fact_6973_relation__of__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),A5: set(A)] : order_relation_of(A,P,A5) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_alh(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),P),A5))) ).

% relation_of_def
tff(fact_6974_size__diff__se,axiom,
    ! [A: $tType,T6: A,S: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,T6),aa(multiset(A),set(A),set_mset(A),S)))
     => ( aa(multiset(A),nat,size_size(multiset(A)),S) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T6),zero_zero(multiset(A)))))),one_one(nat)) ) ) ).

% size_diff_se
tff(fact_6975_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_6976_set__mset__eq__empty__iff,axiom,
    ! [A: $tType,M6: multiset(A)] :
      ( ( aa(multiset(A),set(A),set_mset(A),M6) = bot_bot(set(A)) )
    <=> ( M6 = zero_zero(multiset(A)) ) ) ).

% set_mset_eq_empty_iff
tff(fact_6977_set__mset__union,axiom,
    ! [A: $tType,M6: 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)),plus_plus(multiset(A)),M6),N7)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(multiset(A),set(A),set_mset(A),M6)),aa(multiset(A),set(A),set_mset(A),N7)) ).

% set_mset_union
tff(fact_6978_mset__diff__cancel1elem,axiom,
    ! [A: $tType,A3: A,B4: multiset(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B4)))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),B4) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ).

% mset_diff_cancel1elem
tff(fact_6979_set__mset__Union__mset,axiom,
    ! [A: $tType,MM: multiset(multiset(A))] : aa(multiset(A),set(A),set_mset(A),comm_m7189776963980413722m_mset(multiset(A),MM)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),aa(multiset(multiset(A)),set(multiset(A)),set_mset(multiset(A)),MM))) ).

% set_mset_Union_mset
tff(fact_6980_set__mset__mono,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(multiset(A),set(A),set_mset(A),A5)),aa(multiset(A),set(A),set_mset(A),B4))) ) ).

% set_mset_mono
tff(fact_6981_image__mset__cong,axiom,
    ! [B: $tType,A: $tType,M6: multiset(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(multiset(A),set(A),set_mset(A),M6)))
         => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) )
     => ( aa(multiset(A),multiset(B),image_mset(A,B,F3),M6) = aa(multiset(A),multiset(B),image_mset(A,B,G3),M6) ) ) ).

% image_mset_cong
tff(fact_6982_in__image__mset,axiom,
    ! [A: $tType,B: $tType,Y: A,F3: fun(B,A),M6: multiset(B)] :
      ( pp(aa(set(A),bool,member(A,Y),aa(multiset(A),set(A),set_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),M6))))
    <=> pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),image2(B,A,F3),aa(multiset(B),set(B),set_mset(B),M6)))) ) ).

% in_image_mset
tff(fact_6983_image__mset__cong__pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,M6: multiset(product_prod(A,B)),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C))] :
      ( ! [X3: A,Y3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),M6)))
         => ( aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(B,C,aa(A,fun(B,C),G3,X3),Y3) ) )
     => ( aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)),M6) = aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3)),M6) ) ) ).

% image_mset_cong_pair
tff(fact_6984_in__Inf__multiset__iff,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: A] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5))))
      <=> ! [X4: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),A5))
           => pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),X4))) ) ) ) ).

% in_Inf_multiset_iff
tff(fact_6985_mset__right__cancel__union,axiom,
    ! [A: $tType,A3: A,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(set(A),bool,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)),A5),B4))))
     => ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B4)))
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A5))) ) ) ).

% mset_right_cancel_union
tff(fact_6986_mset__left__cancel__union,axiom,
    ! [A: $tType,A3: A,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(set(A),bool,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)),A5),B4))))
     => ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A5)))
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B4))) ) ) ).

% mset_left_cancel_union
tff(fact_6987_mset__un__cases,axiom,
    ! [A: $tType,A3: A,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(set(A),bool,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)),A5),B4))))
     => ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A5)))
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B4))) ) ) ).

% mset_un_cases
tff(fact_6988_ex__Melem__conv,axiom,
    ! [A: $tType,A5: multiset(A)] :
      ( ? [X4: A] : pp(aa(set(A),bool,member(A,X4),aa(multiset(A),set(A),set_mset(A),A5)))
    <=> ( A5 != zero_zero(multiset(A)) ) ) ).

% ex_Melem_conv
tff(fact_6989_multiset__induct__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(multiset(A),bool),M6: multiset(A)] :
          ( pp(aa(multiset(A),bool,P,zero_zero(multiset(A))))
         => ( ! [X3: A,M11: multiset(A)] :
                ( pp(aa(multiset(A),bool,P,M11))
               => ( ! [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),aa(multiset(A),set(A),set_mset(A),M11)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3)) )
                 => pp(aa(multiset(A),bool,P,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X3),M11))) ) )
           => pp(aa(multiset(A),bool,P,M6)) ) ) ) ).

% multiset_induct_min
tff(fact_6990_multiset__induct__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(multiset(A),bool),M6: multiset(A)] :
          ( pp(aa(multiset(A),bool,P,zero_zero(multiset(A))))
         => ( ! [X3: A,M11: multiset(A)] :
                ( pp(aa(multiset(A),bool,P,M11))
               => ( ! [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),aa(multiset(A),set(A),set_mset(A),M11)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa3),X3)) )
                 => pp(aa(multiset(A),bool,P,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X3),M11))) ) )
           => pp(aa(multiset(A),bool,P,M6)) ) ) ) ).

% multiset_induct_max
tff(fact_6991_mset__right__cancel__elem,axiom,
    ! [A: $tType,A3: A,A5: multiset(A),B2: A] :
      ( pp(aa(set(A),bool,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)),A5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))))))
     => ( ( A3 != B2 )
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A5))) ) ) ).

% mset_right_cancel_elem
tff(fact_6992_mset__left__cancel__elem,axiom,
    ! [A: $tType,A3: A,B2: A,A5: multiset(A)] :
      ( pp(aa(set(A),bool,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)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))),A5))))
     => ( ( A3 != B2 )
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A5))) ) ) ).

% mset_left_cancel_elem
tff(fact_6993_sum__mset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: multiset(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),aa(multiset(B),set(B),set_mset(B),K5)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F3),K5))),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),K5)))) ) ) ).

% sum_mset_mono
tff(fact_6994_set__mset__Inf,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),A5)) ) ) ).

% set_mset_Inf
tff(fact_6995_set__mset__single,axiom,
    ! [A: $tType,B2: A] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) ).

% set_mset_single
tff(fact_6996_mset__2dist2__cases,axiom,
    ! [A: $tType,A3: A,B2: A,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4)))
     => ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),A5))
       => ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),B4))
         => ( ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A5)))
             => ~ pp(aa(set(A),bool,member(A,B2),aa(multiset(A),set(A),set_mset(A),B4))) )
           => ~ ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B4)))
               => ~ pp(aa(set(A),bool,member(A,B2),aa(multiset(A),set(A),set_mset(A),A5))) ) ) ) ) ) ).

% mset_2dist2_cases
tff(fact_6997_mset__union__subset__s,axiom,
    ! [A: $tType,A3: A,B4: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),B4)),C4))
     => ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),C4)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B4),C4)) ) ) ).

% mset_union_subset_s
tff(fact_6998_mset__le__mono__add__single,axiom,
    ! [A: $tType,A3: A,Ys2: multiset(A),B2: A,Ws2: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),Ys2)))
     => ( pp(aa(set(A),bool,member(A,B2),aa(multiset(A),set(A),set_mset(A),Ws2)))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ys2),Ws2))) ) ) ).

% mset_le_mono_add_single
tff(fact_6999_nth__mem__mset,axiom,
    ! [A: $tType,I2: nat,Ls2: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ls2)))
     => pp(aa(set(A),bool,member(A,aa(nat,A,nth(A,Ls2),I2)),aa(multiset(A),set(A),set_mset(A),mset(A,Ls2)))) ) ).

% nth_mem_mset
tff(fact_7000_mset__un__single__un__cases,axiom,
    ! [A: $tType,A3: A,A5: multiset(A),B4: multiset(A),C4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),C4) )
     => ( ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B4)))
         => ( A5 != 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)),B4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))),C4) ) )
       => ~ ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),C4)))
           => ( A5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))) ) ) ) ) ).

% mset_un_single_un_cases
tff(fact_7001_diff__union__single__conv2,axiom,
    ! [A: $tType,A3: A,J5: multiset(A),I5: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),J5)))
     => ( 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)),J5),I5)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),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)),J5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))),I5) ) ) ).

% diff_union_single_conv2
tff(fact_7002_mset__union__diff__comm,axiom,
    ! [A: $tType,T6: A,S: multiset(A),T2: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,T6),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)),T2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T6),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)),T2),S)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T6),zero_zero(multiset(A)))) ) ) ).

% mset_union_diff_comm
tff(fact_7003_mset__contains__eq,axiom,
    ! [A: $tType,M: A,M6: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,M),aa(multiset(A),set(A),set_mset(A),M6)))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M),zero_zero(multiset(A))))) = M6 ) ) ).

% mset_contains_eq
tff(fact_7004_mset__size1elem,axiom,
    ! [A: $tType,P: multiset(A),Q3: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),one_one(nat)))
     => ( pp(aa(set(A),bool,member(A,Q3),aa(multiset(A),set(A),set_mset(A),P)))
       => ( P = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q3),zero_zero(multiset(A))) ) ) ) ).

% mset_size1elem
tff(fact_7005_size__Diff1__less,axiom,
    ! [A: $tType,X: A,M6: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ) ).

% size_Diff1_less
tff(fact_7006_size__Diff2__less,axiom,
    ! [A: $tType,X: A,M6: multiset(A),Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6)))
     => ( pp(aa(set(A),bool,member(A,Y),aa(multiset(A),set(A),set_mset(A),M6)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Y),zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ) ) ).

% size_Diff2_less
tff(fact_7007_at__most__one__mset__mset__diff,axiom,
    ! [A: $tType,A3: A,M6: multiset(A)] :
      ( ~ pp(aa(set(A),bool,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)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))))))
     => ( aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(multiset(A),set(A),set_mset(A),M6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ) ).

% at_most_one_mset_mset_diff
tff(fact_7008_mult1__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : mult1(A,R3) = aa(fun(product_prod(multiset(A),multiset(A)),bool),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),bool)),fun(product_prod(multiset(A),multiset(A)),bool),product_case_prod(multiset(A),multiset(A),bool),aTP_Lamp_ali(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),bool)),R3))) ).

% mult1_def
tff(fact_7009_sum__wcount__Int,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat),N7: multiset(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F3,N7)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(multiset(A),set(A),set_mset(A),N7))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F3,N7)),A5) ) ) ).

% sum_wcount_Int
tff(fact_7010_not__less__empty,axiom,
    ! [A: $tType,M6: multiset(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),M6),zero_zero(multiset(A)))),mult1(A,R3))) ).

% not_less_empty
tff(fact_7011_mult1__union,axiom,
    ! [A: $tType,B4: multiset(A),D4: multiset(A),R3: set(product_prod(A,A)),C4: multiset(A)] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),B4),D4)),mult1(A,R3)))
     => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),C4),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D4))),mult1(A,R3))) ) ).

% mult1_union
tff(fact_7012_mono__mult1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R6))
     => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),bool),ord_less_eq(set(product_prod(multiset(A),multiset(A)))),mult1(A,R3)),mult1(A,R6))) ) ).

% mono_mult1
tff(fact_7013_less__add,axiom,
    ! [A: $tType,N7: multiset(A),A3: A,M0: multiset(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M0))),mult1(A,R3)))
     => ( ? [M11: multiset(A)] :
            ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),M11),M0)),mult1(A,R3)))
            & ( N7 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M11) ) )
        | ? [K8: multiset(A)] :
            ( ! [B11: A] :
                ( pp(aa(set(A),bool,member(A,B11),aa(multiset(A),set(A),set_mset(A),K8)))
               => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B11),A3)),R3)) )
            & ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K8) ) ) ) ) ).

% less_add
tff(fact_7014_mult1I,axiom,
    ! [A: $tType,M6: multiset(A),A3: A,M0: multiset(A),N7: multiset(A),K5: multiset(A),R3: set(product_prod(A,A))] :
      ( ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M0) )
     => ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K5) )
       => ( ! [B5: A] :
              ( pp(aa(set(A),bool,member(A,B5),aa(multiset(A),set(A),set_mset(A),K5)))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A3)),R3)) )
         => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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),M6)),mult1(A,R3))) ) ) ) ).

% mult1I
tff(fact_7015_mult1E,axiom,
    ! [A: $tType,N7: multiset(A),M6: multiset(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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),M6)),mult1(A,R3)))
     => ~ ! [A6: A,M02: multiset(A)] :
            ( ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A6),M02) )
           => ! [K8: multiset(A)] :
                ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K8) )
               => ~ ! [B11: A] :
                      ( pp(aa(set(A),bool,member(A,B11),aa(multiset(A),set(A),set_mset(A),K8)))
                     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B11),A6)),R3)) ) ) ) ) ).

% mult1E
tff(fact_7016_mult1__lessE,axiom,
    ! [A: $tType,N7: multiset(A),M6: multiset(A),R3: fun(A,fun(A,bool))] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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),M6)),mult1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3)))))
     => ( asymp(A,R3)
       => ~ ! [A6: A,M02: multiset(A)] :
              ( ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A6),M02) )
             => ! [K8: multiset(A)] :
                  ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K8) )
                 => ( ~ pp(aa(set(A),bool,member(A,A6),aa(multiset(A),set(A),set_mset(A),K8)))
                   => ~ ! [B11: A] :
                          ( pp(aa(set(A),bool,member(A,B11),aa(multiset(A),set(A),set_mset(A),K8)))
                         => pp(aa(A,bool,aa(A,fun(A,bool),R3,B11),A6)) ) ) ) ) ) ) ).

% mult1_lessE
tff(fact_7017_mult__implies__one__step,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),M6: multiset(A),N7: multiset(A)] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),M6),N7)),mult(A,R3)))
       => ? [I7: multiset(A),J6: multiset(A)] :
            ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),J6) )
            & ? [K8: multiset(A)] :
                ( ( M6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),K8) )
                & ( J6 != zero_zero(multiset(A)) )
                & ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),aa(multiset(A),set(A),set_mset(A),K8)))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),aa(multiset(A),set(A),set_mset(A),J6)))
                        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa4)),R3)) ) ) ) ) ) ) ).

% mult_implies_one_step
tff(fact_7018_Multiset_Omono__mult,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R6))
     => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),bool),ord_less_eq(set(product_prod(multiset(A),multiset(A)))),mult(A,R3)),mult(A,R6))) ) ).

% Multiset.mono_mult
tff(fact_7019_subset__implies__mult,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A),R3: set(product_prod(A,A))] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A5),B4))
     => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),A5),B4)),mult(A,R3))) ) ).

% subset_implies_mult
tff(fact_7020_asymp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,aTP_Lamp_bo(A,fun(A,bool))) ) ).

% asymp_greater
tff(fact_7021_asympD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A,Y: A] :
      ( asymp(A,R2)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y))
       => ~ pp(aa(A,bool,aa(A,fun(A,bool),R2,Y),X)) ) ) ).

% asympD
tff(fact_7022_asymp_Ointros,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( ! [A6: A,B5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,A6),B5))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),R2,B5),A6)) )
     => asymp(A,R2) ) ).

% asymp.intros
tff(fact_7023_asymp_Osimps,axiom,
    ! [A: $tType,A3: fun(A,fun(A,bool))] :
      ( asymp(A,A3)
    <=> ? [R10: fun(A,fun(A,bool))] :
          ( ( A3 = R10 )
          & ! [X4: A,Xa: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),R10,X4),Xa))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),R10,Xa),X4)) ) ) ) ).

% asymp.simps
tff(fact_7024_asymp_Ocases,axiom,
    ! [A: $tType,A3: fun(A,fun(A,bool))] :
      ( asymp(A,A3)
     => ! [A14: A,B11: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),A3,A14),B11))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),A3,B11),A14)) ) ) ).

% asymp.cases
tff(fact_7025_asymp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,ord_less(A)) ) ).

% asymp_less
tff(fact_7026_subset__mset_Oasymp__greater,axiom,
    ! [A: $tType] : asymp(multiset(A),aTP_Lamp_acv(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.asymp_greater
tff(fact_7027_mult__cancel__add__mset,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),Uu: A,X6: multiset(A),Y6: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),X6)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),Y6))),mult(A,S2)))
        <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),X6),Y6)),mult(A,S2))) ) ) ) ).

% mult_cancel_add_mset
tff(fact_7028_mult__cancel,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X6: multiset(A),Z4: multiset(A),Y6: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),X6),Z4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y6),Z4))),mult(A,S2)))
        <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),X6),Y6)),mult(A,S2))) ) ) ) ).

% mult_cancel
tff(fact_7029_mult__cancel__max,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X6: multiset(A),Y6: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),X6),Y6)),mult(A,S2)))
        <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),X6),Y6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y6),X6))),mult(A,S2))) ) ) ) ).

% mult_cancel_max
tff(fact_7030_one__step__implies__mult,axiom,
    ! [A: $tType,J5: multiset(A),K5: multiset(A),R3: set(product_prod(A,A)),I5: multiset(A)] :
      ( ( J5 != zero_zero(multiset(A)) )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(multiset(A),set(A),set_mset(A),K5)))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),aa(multiset(A),set(A),set_mset(A),J5)))
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R3)) ) )
       => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),I5),K5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),J5))),mult(A,R3))) ) ) ).

% one_step_implies_mult
tff(fact_7031_multp__code__iff__mult,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( irrefl(A,R2)
     => ( trans(A,R2)
       => ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y3))
            <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R2)) )
         => ( multp_code(A,P,N7,M6)
          <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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),M6)),mult(A,R2))) ) ) ) ) ).

% multp_code_iff_mult
tff(fact_7032_multeqp__code__iff__reflcl__mult,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( irrefl(A,R2)
     => ( trans(A,R2)
       => ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y3))
            <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R2)) )
         => ( multeqp_code(A,P,N7,M6)
          <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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),M6)),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,R2)),id2(multiset(A))))) ) ) ) ) ).

% multeqp_code_iff_reflcl_mult
tff(fact_7033_multiset_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: multiset(A),B2: multiset(B)] :
      ( pp(aa(multiset(B),bool,aa(multiset(A),fun(multiset(B),bool),rel_mset(A,B,R2),A3),B2))
    <=> ? [Z6: multiset(product_prod(A,B))] :
          ( pp(aa(set(multiset(product_prod(A,B))),bool,member(multiset(product_prod(A,B)),Z6),aa(fun(multiset(product_prod(A,B)),bool),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_alj(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),R2))))
          & ( aa(multiset(product_prod(A,B)),multiset(A),image_mset(product_prod(A,B),A,product_fst(A,B)),Z6) = A3 )
          & ( aa(multiset(product_prod(A,B)),multiset(B),image_mset(product_prod(A,B),B,product_snd(A,B)),Z6) = B2 ) ) ) ).

% multiset.in_rel
tff(fact_7034_multp__code__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( multp_code(A,P,N7,M6)
    <=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)) != zero_zero(multiset(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),N7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
           => ? [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),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)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
                & pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Xa)) ) ) ) ) ).

% multp_code_def
tff(fact_7035_set__mset__inter,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(multiset(A),set(A),set_mset(A),A5)),aa(multiset(A),set(A),set_mset(A),B4)) ).

% set_mset_inter
tff(fact_7036_multiset_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: multiset(A),G3: fun(B,C),Y: multiset(B)] :
      ( pp(aa(multiset(C),bool,aa(multiset(A),fun(multiset(C),bool),rel_mset(A,C,Sa),X),aa(multiset(B),multiset(C),image_mset(B,C,G3),Y)))
    <=> pp(aa(multiset(B),bool,aa(multiset(A),fun(multiset(B),bool),rel_mset(A,B,aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahi(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G3)),X),Y)) ) ).

% multiset.rel_map(2)
tff(fact_7037_multiset_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: multiset(A),Y: multiset(B)] :
      ( pp(aa(multiset(B),bool,aa(multiset(C),fun(multiset(B),bool),rel_mset(C,B,Sb),aa(multiset(A),multiset(C),image_mset(A,C,I2),X)),Y))
    <=> pp(aa(multiset(B),bool,aa(multiset(A),fun(multiset(B),bool),rel_mset(A,B,aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% multiset.rel_map(1)
tff(fact_7038_multiset_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra2))
     => pp(aa(fun(multiset(A),fun(multiset(B),bool)),bool,aa(fun(multiset(A),fun(multiset(B),bool)),fun(fun(multiset(A),fun(multiset(B),bool)),bool),ord_less_eq(fun(multiset(A),fun(multiset(B),bool))),rel_mset(A,B,R2)),rel_mset(A,B,Ra2))) ) ).

% multiset.rel_mono
tff(fact_7039_mult__cancel__max0,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X6: multiset(A),Y6: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),X6),Y6)),mult(A,S2)))
        <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),X6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X6),Y6))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X6),Y6)))),mult(A,S2))) ) ) ) ).

% mult_cancel_max0
tff(fact_7040_multeqp__code__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( multeqp_code(A,P,N7,M6)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),N7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
         => ? [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),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)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
              & pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Xa)) ) ) ) ).

% multeqp_code_def
tff(fact_7041_count__image__mset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: multiset(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),A5)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),count(B),A5)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),aa(multiset(B),set(B),set_mset(B),A5))) ).

% count_image_mset
tff(fact_7042_set__mset__replicate__mset__subset,axiom,
    ! [A: $tType,N2: nat,X: A] :
      ( ( ( N2 = zero_zero(nat) )
       => ( aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N2,X)) = bot_bot(set(A)) ) )
      & ( ( N2 != zero_zero(nat) )
       => ( aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N2,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ).

% set_mset_replicate_mset_subset
tff(fact_7043_mset__empty__count,axiom,
    ! [A: $tType,M6: multiset(A)] :
      ( ! [P4: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),P4) = zero_zero(nat)
    <=> ( M6 = zero_zero(multiset(A)) ) ) ).

% mset_empty_count
tff(fact_7044_count__greater__zero__iff,axiom,
    ! [A: $tType,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6))) ) ).

% count_greater_zero_iff
tff(fact_7045_count__greater__eq__one__iff,axiom,
    ! [A: $tType,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6))) ) ).

% count_greater_eq_one_iff
tff(fact_7046_in__replicate__mset,axiom,
    ! [A: $tType,X: A,N2: nat,Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N2,Y))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
        & ( X = Y ) ) ) ).

% in_replicate_mset
tff(fact_7047_sum__mset__replicate__mset,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N2: nat,A3: A] : comm_m7189776963980413722m_mset(A,replicate_mset(A,N2,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),A3) ) ).

% sum_mset_replicate_mset
tff(fact_7048_count__greater__eq__Suc__zero__iff,axiom,
    ! [A: $tType,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6))) ) ).

% count_greater_eq_Suc_zero_iff
tff(fact_7049_msubseteq__replicate__msetE,axiom,
    ! [A: $tType,A5: multiset(A),N2: nat,A3: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),replicate_mset(A,N2,A3)))
     => ~ ! [M5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N2))
           => ( A5 != replicate_mset(A,M5,A3) ) ) ) ).

% msubseteq_replicate_msetE
tff(fact_7050_count__le__replicate__mset__subset__eq,axiom,
    ! [A: $tType,N2: nat,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),replicate_mset(A,N2,X)),M6)) ) ).

% count_le_replicate_mset_subset_eq
tff(fact_7051_mset__subset__eqI,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A)] :
      ( ! [A6: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),A5),A6)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),B4),A6)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),B4)) ) ).

% mset_subset_eqI
tff(fact_7052_subseteq__mset__def,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),B4))
    <=> ! [A8: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),A5),A8)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),B4),A8))) ) ).

% subseteq_mset_def
tff(fact_7053_mset__subset__eq__count,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A),A3: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A5),B4))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),A5),A3)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),B4),A3))) ) ).

% mset_subset_eq_count
tff(fact_7054_count__sum,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),A5: set(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(B),multiset(A),aa(fun(B,multiset(A)),fun(set(B),multiset(A)),groups7311177749621191930dd_sum(B,multiset(A)),F3),A5)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(A,fun(B,nat),aTP_Lamp_alk(fun(B,multiset(A)),fun(A,fun(B,nat)),F3),X)),A5) ).

% count_sum
tff(fact_7055_in__diff__count,axiom,
    ! [A: $tType,A3: A,M6: multiset(A),N7: multiset(A)] :
      ( pp(aa(set(A),bool,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)),M6),N7))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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),M6),A3))) ) ).

% in_diff_count
tff(fact_7056_count__ne__remove,axiom,
    ! [A: $tType,X: A,T6: A,S: multiset(A)] :
      ( ( X != T6 )
     => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),S),X) = 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),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T6),zero_zero(multiset(A))))),X) ) ) ).

% count_ne_remove
tff(fact_7057_set__mset__def,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),set(A),set_mset(A),M6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_all(multiset(A),fun(A,bool),M6)) ).

% set_mset_def
tff(fact_7058_set__mset__diff,axiom,
    ! [A: $tType,M6: 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)),M6),N7)) = aa(fun(A,bool),set(A),collect(A),aa(multiset(A),fun(A,bool),aTP_Lamp_alm(multiset(A),fun(multiset(A),fun(A,bool)),M6),N7)) ).

% set_mset_diff
tff(fact_7059_count__mset,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset(A,Xs)),X) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),X)),Xs)) ).

% count_mset
tff(fact_7060_count__image__mset_H,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),X6: multiset(B),Y: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),X6)),Y) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),count(B),X6)),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aa(multiset(B),fun(A,fun(B,bool)),aTP_Lamp_aln(fun(B,A),fun(multiset(B),fun(A,fun(B,bool))),F3),X6),Y))) ).

% count_image_mset'
tff(fact_7061_size__multiset__overloaded__eq,axiom,
    ! [B: $tType,X: multiset(B)] : aa(multiset(B),nat,size_size(multiset(B)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),count(B),X)),aa(multiset(B),set(B),set_mset(B),X)) ).

% size_multiset_overloaded_eq
tff(fact_7062_count__induct,axiom,
    ! [A: $tType,Y: fun(A,nat),P: fun(fun(A,nat),bool)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool))))
     => ( ! [X3: multiset(A)] : pp(aa(fun(A,nat),bool,P,aa(multiset(A),fun(A,nat),count(A),X3)))
       => pp(aa(fun(A,nat),bool,P,Y)) ) ) ).

% count_induct
tff(fact_7063_count__cases,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool))))
     => ~ ! [X3: multiset(A)] : Y != aa(multiset(A),fun(A,nat),count(A),X3) ) ).

% count_cases
tff(fact_7064_count,axiom,
    ! [A: $tType,X: multiset(A)] : pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),aa(multiset(A),fun(A,nat),count(A),X)),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool)))) ).

% count
tff(fact_7065_image__mset__const__eq,axiom,
    ! [B: $tType,A: $tType,C2: A,M6: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_px(A,fun(B,A)),C2)),M6) = replicate_mset(A,aa(multiset(B),nat,size_size(multiset(B)),M6),C2) ).

% image_mset_const_eq
tff(fact_7066_Inf__multiset_Orep__eq,axiom,
    ! [A: $tType,X: set(multiset(A)),X5: A] :
      ( ( ( aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X) = bot_bot(set(fun(A,nat))) )
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X)),X5) = zero_zero(nat) ) )
      & ( ( aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X) != bot_bot(set(fun(A,nat))) )
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X)),X5) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_alp(A,fun(fun(A,nat),nat),X5)),aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X))) ) ) ) ).

% Inf_multiset.rep_eq
tff(fact_7067_count__Inf__multiset__nonempty,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: A] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5)),X) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_alq(A,fun(multiset(A),nat),X)),A5)) ) ) ).

% count_Inf_multiset_nonempty
tff(fact_7068_count__mset__gt__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset(A,Xs)),X))) ) ).

% count_mset_gt_0
tff(fact_7069_replicate__mset__msubseteq__iff,axiom,
    ! [A: $tType,M: nat,A3: A,N2: nat,B2: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),replicate_mset(A,M,A3)),replicate_mset(A,N2,B2)))
    <=> ( ( M = zero_zero(nat) )
        | ( ( A3 = B2 )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2)) ) ) ) ).

% replicate_mset_msubseteq_iff
tff(fact_7070_sum__mset__delta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C2: A,A5: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_alr(B,fun(A,fun(B,A)),Y),C2)),A5)) = 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),A5),Y))) ) ).

% sum_mset_delta'
tff(fact_7071_sum__mset__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C2: A,A5: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_als(B,fun(A,fun(B,A)),Y),C2)),A5)) = 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),A5),Y))) ) ).

% sum_mset_delta
tff(fact_7072_bdd__above__multiset__imp__finite__support,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
       => pp(aa(set(A),bool,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_alt(multiset(A),set(A))),A5)))) ) ) ).

% bdd_above_multiset_imp_finite_support
tff(fact_7073_size__multiset__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),M6: multiset(A)] : aa(multiset(A),nat,size_multiset(A,F3),M6) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(multiset(A),fun(A,nat),aTP_Lamp_alu(fun(A,nat),fun(multiset(A),fun(A,nat)),F3),M6)),aa(multiset(A),set(A),set_mset(A),M6)) ).

% size_multiset_eq
tff(fact_7074_subset__mset_Obdd__above__Un,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A5),B4))
    <=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
        & condit8047198070973881523_above(multiset(A),subseteq_mset(A),B4) ) ) ).

% subset_mset.bdd_above_Un
tff(fact_7075_subset__mset_Obdd__above__empty,axiom,
    ! [A: $tType] : condit8047198070973881523_above(multiset(A),subseteq_mset(A),bot_bot(set(multiset(A)))) ).

% subset_mset.bdd_above_empty
tff(fact_7076_subset__mset_Obdd__above__UN,axiom,
    ! [A: $tType,B: $tType,I5: set(B),A5: fun(B,set(multiset(A)))] :
      ( pp(aa(set(B),bool,finite_finite2(B),I5))
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(set(multiset(A))),set(multiset(A)),complete_Sup_Sup(set(multiset(A))),aa(set(B),set(set(multiset(A))),image2(B,set(multiset(A)),A5),I5)))
      <=> ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),I5))
           => condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(B,set(multiset(A)),A5,X4)) ) ) ) ).

% subset_mset.bdd_above_UN
tff(fact_7077_subset__mset_Obdd__above__mono,axiom,
    ! [A: $tType,B4: set(multiset(A)),A5: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B4)
     => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A5),B4))
       => condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5) ) ) ).

% subset_mset.bdd_above_mono
tff(fact_7078_subset__mset_Obdd__above__Int2,axiom,
    ! [A: $tType,B4: set(multiset(A)),A5: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B4)
     => condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4)) ) ).

% subset_mset.bdd_above_Int2
tff(fact_7079_subset__mset_Obdd__above__Int1,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
     => condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4)) ) ).

% subset_mset.bdd_above_Int1
tff(fact_7080_subset__mset_OcSup__mono,axiom,
    ! [A: $tType,B4: set(multiset(A)),A5: set(multiset(A))] :
      ( ( B4 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
       => ( ! [B5: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),B5),B4))
             => ? [X5: multiset(A)] :
                  ( pp(aa(set(multiset(A)),bool,member(multiset(A),X5),A5))
                  & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),X5)) ) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B4)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5))) ) ) ) ).

% subset_mset.cSup_mono
tff(fact_7081_subset__mset_OcSup__le__iff,axiom,
    ! [A: $tType,S: set(multiset(A)),A3: multiset(A)] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S)),A3))
        <=> ! [X4: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),S))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X4),A3)) ) ) ) ) ).

% subset_mset.cSup_le_iff
tff(fact_7082_size__multiset__overloaded__def,axiom,
    ! [B: $tType] : size_size(multiset(B)) = size_multiset(B,aTP_Lamp_alv(B,nat)) ).

% size_multiset_overloaded_def
tff(fact_7083_subset__mset_OcSUP__le__iff,axiom,
    ! [B: $tType,A: $tType,A5: set(B),F3: fun(B,multiset(A)),U: multiset(A)] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),U))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,X4)),U)) ) ) ) ) ).

% subset_mset.cSUP_le_iff
tff(fact_7084_subset__mset_OcSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(B),G3: fun(C,multiset(A)),B4: set(C),F3: fun(B,multiset(A))] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),G3),B4))
       => ( ! [N5: B] :
              ( pp(aa(set(B),bool,member(B,N5),A5))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),B4))
                  & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,N5)),aa(C,multiset(A),G3,X5))) ) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),G3),B4)))) ) ) ) ).

% subset_mset.cSUP_mono
tff(fact_7085_subset__mset_OcSup__subset__mono,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B4)
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A5),B4))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B4))) ) ) ) ).

% subset_mset.cSup_subset_mono
tff(fact_7086_subset__mset_OcSUP__lessD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,multiset(A)),A5: set(B),Y: multiset(A),I2: B] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),Y))
       => ( pp(aa(set(B),bool,member(B,I2),A5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(B,multiset(A),F3,I2)),Y)) ) ) ) ).

% subset_mset.cSUP_lessD
tff(fact_7087_subset__mset_OcSup__cInf,axiom,
    ! [A: $tType,S: set(multiset(A))] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aTP_Lamp_alw(set(multiset(A)),fun(multiset(A),bool),S))) ) ) ) ).

% subset_mset.cSup_cInf
tff(fact_7088_subset__mset_OcSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType,A5: set(B),G3: fun(B,multiset(A)),B4: set(B),F3: fun(B,multiset(A))] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B4))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
               => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,X3)),aa(B,multiset(A),G3,X3))) )
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B4)))) ) ) ) ) ).

% subset_mset.cSUP_subset_mono
tff(fact_7089_set__mset__Sup,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
     => ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),A5)) ) ) ).

% set_mset_Sup
tff(fact_7090_count__Sup__multiset__nonempty,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: A] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5)),X) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_alq(A,fun(multiset(A),nat),X)),A5)) ) ) ) ).

% count_Sup_multiset_nonempty
tff(fact_7091_Sup__multiset__in__multiset,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
       => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_alx(set(multiset(A)),fun(A,bool),A5)))) ) ) ).

% Sup_multiset_in_multiset
tff(fact_7092_subset__mset_Omono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A5: fun(C,multiset(A)),I5: set(C)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),A5),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_aly(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F3),A5)),I5))),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A5),I5))))) ) ) ) ) ).

% subset_mset.mono_cSUP
tff(fact_7093_subset__mset_Omono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A5: set(multiset(A))] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
           => ( ( A5 != bot_bot(set(multiset(A))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F3),A5))),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5)))) ) ) ) ) ).

% subset_mset.mono_cSup
tff(fact_7094_order_Omono_Ocong,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [Less_eq: fun(A,fun(A,bool))] : mono(A,B,Less_eq) = mono(A,B,Less_eq) ) ).

% order.mono.cong
tff(fact_7095_subset__mset_OmonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B),X: multiset(A),Y: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X)),aa(multiset(A),B,F3,Y))) ) ) ) ).

% subset_mset.monoD
tff(fact_7096_subset__mset_OmonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B),X: multiset(A),Y: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X)),aa(multiset(A),B,F3,Y))) ) ) ) ).

% subset_mset.monoE
tff(fact_7097_subset__mset_OmonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B)] :
          ( ! [X3: multiset(A),Y3: multiset(A)] :
              ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X3)),aa(multiset(A),B,F3,Y3))) )
         => pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3)) ) ) ).

% subset_mset.monoI
tff(fact_7098_subset__mset_Omono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
        <=> ! [X4: multiset(A),Y5: multiset(A)] :
              ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X4)),aa(multiset(A),B,F3,Y5))) ) ) ) ).

% subset_mset.mono_def
tff(fact_7099_subset__mset_Omono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F3: fun(multiset(A),B),A5: multiset(A),B4: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A5),B4))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(multiset(A),B,F3,A5)),aa(multiset(A),B,F3,B4)))) ) ) ).

% subset_mset.mono_inf
tff(fact_7100_Inf__multiset__def,axiom,
    ! [A: $tType] : complete_Inf_Inf(multiset(A)) = aa(fun(set(fun(A,nat)),fun(A,nat)),fun(set(multiset(A)),multiset(A)),map_fun(set(multiset(A)),set(fun(A,nat)),fun(A,nat),multiset(A),image2(multiset(A),fun(A,nat),count(A)),abs_multiset(A)),aTP_Lamp_alz(set(fun(A,nat)),fun(A,nat))) ).

% Inf_multiset_def
tff(fact_7101_Sup__multiset__def,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( ( ( ( A5 != bot_bot(set(multiset(A))) )
          & condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5) )
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5) = aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_ama(set(multiset(A)),fun(A,nat),A5)) ) )
      & ( ~ ( ( A5 != bot_bot(set(multiset(A))) )
            & condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5) )
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5) = zero_zero(multiset(A)) ) ) ) ).

% Sup_multiset_def
tff(fact_7102_count__Abs__multiset,axiom,
    ! [A: $tType,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_gi(fun(A,nat),fun(A,bool),F3))))
     => ( aa(multiset(A),fun(A,nat),count(A),aa(fun(A,nat),multiset(A),abs_multiset(A),F3)) = F3 ) ) ).

% count_Abs_multiset
tff(fact_7103_zero__multiset__def,axiom,
    ! [A: $tType] : zero_zero(multiset(A)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_amb(A,nat)) ).

% zero_multiset_def
tff(fact_7104_Abs__multiset__cases,axiom,
    ! [A: $tType,X: multiset(A)] :
      ~ ! [Y3: fun(A,nat)] :
          ( ( X = aa(fun(A,nat),multiset(A),abs_multiset(A),Y3) )
         => ~ pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y3),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool)))) ) ).

% Abs_multiset_cases
tff(fact_7105_Abs__multiset__induct,axiom,
    ! [A: $tType,P: fun(multiset(A),bool),X: multiset(A)] :
      ( ! [Y3: fun(A,nat)] :
          ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y3),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool))))
         => pp(aa(multiset(A),bool,P,aa(fun(A,nat),multiset(A),abs_multiset(A),Y3))) )
     => pp(aa(multiset(A),bool,P,X)) ) ).

% Abs_multiset_induct
tff(fact_7106_Abs__multiset__inject,axiom,
    ! [A: $tType,X: fun(A,nat),Y: fun(A,nat)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),X),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool))))
     => ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool))))
       => ( ( aa(fun(A,nat),multiset(A),abs_multiset(A),X) = aa(fun(A,nat),multiset(A),abs_multiset(A),Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_multiset_inject
tff(fact_7107_plus__multiset__def,axiom,
    ! [A: $tType] : plus_plus(multiset(A)) = aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(multiset(A),fun(multiset(A),multiset(A))),map_fun(multiset(A),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),count(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_amc(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% plus_multiset_def
tff(fact_7108_minus__multiset__def,axiom,
    ! [A: $tType] : minus_minus(multiset(A)) = aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(multiset(A),fun(multiset(A),multiset(A))),map_fun(multiset(A),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),count(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_amd(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% minus_multiset_def
tff(fact_7109_Abs__multiset__inverse,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool))))
     => ( aa(multiset(A),fun(A,nat),count(A),aa(fun(A,nat),multiset(A),abs_multiset(A),Y)) = Y ) ) ).

% Abs_multiset_inverse
tff(fact_7110_add__mset__def,axiom,
    ! [A: $tType] : add_mset(A) = aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(A,fun(multiset(A),multiset(A))),map_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_ame(A,fun(fun(A,nat),fun(A,nat)))) ).

% add_mset_def
tff(fact_7111_subset__mset_Omono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A5: fun(C,multiset(A)),I5: set(C)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),A5),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A5),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_aly(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F3),A5)),I5)))) ) ) ) ) ).

% subset_mset.mono_cINF
tff(fact_7112_subset__mset_Omono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A5: set(multiset(A))] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5)
           => ( ( A5 != bot_bot(set(multiset(A))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F3),A5)))) ) ) ) ) ).

% subset_mset.mono_cInf
tff(fact_7113_subset__mset_Obdd__below__empty,axiom,
    ! [A: $tType] : condit8119078960628432327_below(multiset(A),subseteq_mset(A),bot_bot(set(multiset(A)))) ).

% subset_mset.bdd_below_empty
tff(fact_7114_subset__mset_Obdd__below__Un,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A5),B4))
    <=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5)
        & condit8119078960628432327_below(multiset(A),subseteq_mset(A),B4) ) ) ).

% subset_mset.bdd_below_Un
tff(fact_7115_subset__mset_Obdd__below__image__inf,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),G3: fun(B,multiset(A)),A5: set(B)] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_amf(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A5))
    <=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
        & condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A5)) ) ) ).

% subset_mset.bdd_below_image_inf
tff(fact_7116_subset__mset_Obdd__below__UN,axiom,
    ! [A: $tType,B: $tType,I5: set(B),A5: fun(B,set(multiset(A)))] :
      ( pp(aa(set(B),bool,finite_finite2(B),I5))
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(set(multiset(A))),set(multiset(A)),complete_Sup_Sup(set(multiset(A))),aa(set(B),set(set(multiset(A))),image2(B,set(multiset(A)),A5),I5)))
      <=> ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),I5))
           => condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(B,set(multiset(A)),A5,X4)) ) ) ) ).

% subset_mset.bdd_below_UN
tff(fact_7117_subset__mset_Obdd__below__Int1,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5)
     => condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4)) ) ).

% subset_mset.bdd_below_Int1
tff(fact_7118_subset__mset_Obdd__below__Int2,axiom,
    ! [A: $tType,B4: set(multiset(A)),A5: set(multiset(A))] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B4)
     => condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4)) ) ).

% subset_mset.bdd_below_Int2
tff(fact_7119_subset__mset_Obdd__below__mono,axiom,
    ! [A: $tType,B4: set(multiset(A)),A5: set(multiset(A))] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B4)
     => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A5),B4))
       => condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5) ) ) ).

% subset_mset.bdd_below_mono
tff(fact_7120_subset__mset_Ole__cInf__iff,axiom,
    ! [A: $tType,S: set(multiset(A)),A3: multiset(A)] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),S)
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S)))
        <=> ! [X4: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),S))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),X4)) ) ) ) ) ).

% subset_mset.le_cInf_iff
tff(fact_7121_subset__mset_OcInf__mono,axiom,
    ! [A: $tType,B4: set(multiset(A)),A5: set(multiset(A))] :
      ( ( B4 != bot_bot(set(multiset(A))) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5)
       => ( ! [B5: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),B5),B4))
             => ? [X5: multiset(A)] :
                  ( pp(aa(set(multiset(A)),bool,member(multiset(A),X5),A5))
                  & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X5),B5)) ) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B4))) ) ) ) ).

% subset_mset.cInf_mono
tff(fact_7122_subset__mset_OcINF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType,B4: set(B),F3: fun(C,multiset(A)),A5: set(C),G3: fun(B,multiset(A))] :
      ( ( B4 != bot_bot(set(B)) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),F3),A5))
       => ( ! [M5: B] :
              ( pp(aa(set(B),bool,member(B,M5),B4))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),A5))
                  & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(C,multiset(A),F3,X5)),aa(B,multiset(A),G3,M5))) ) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),F3),A5))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B4)))) ) ) ) ).

% subset_mset.cINF_mono
tff(fact_7123_subset__mset_Ole__cINF__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),U: multiset(A)] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),U),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),U),aa(B,multiset(A),F3,X4))) ) ) ) ) ).

% subset_mset.le_cINF_iff
tff(fact_7124_subset__mset_OcInf__superset__mono,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B4)
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A5),B4))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B4)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5))) ) ) ) ).

% subset_mset.cInf_superset_mono
tff(fact_7125_subset__mset_Oless__cINF__D,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),A5: set(B),Y: multiset(A),I2: B] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Y),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))))
       => ( pp(aa(set(B),bool,member(B,I2),A5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Y),aa(B,multiset(A),F3,I2))) ) ) ) ).

% subset_mset.less_cINF_D
tff(fact_7126_subset__mset_OcINF__superset__mono,axiom,
    ! [A: $tType,B: $tType,A5: set(B),G3: fun(B,multiset(A)),B4: set(B),F3: fun(B,multiset(A))] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B4))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),B4))
               => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),G3,X3)),aa(B,multiset(A),F3,X3))) )
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B4))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5)))) ) ) ) ) ).

% subset_mset.cINF_superset_mono
tff(fact_7127_subset__mset_OcInf__insert__If,axiom,
    ! [A: $tType,X6: set(multiset(A)),A3: multiset(A)] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),X6)
     => ( ( ( X6 = bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),X6)) = A3 ) )
        & ( ( X6 != bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),X6)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X6)) ) ) ) ) ).

% subset_mset.cInf_insert_If
tff(fact_7128_subset__mset_OcInf__insert,axiom,
    ! [A: $tType,X6: set(multiset(A)),A3: multiset(A)] :
      ( ( X6 != bot_bot(set(multiset(A))) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),X6)
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),X6)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X6)) ) ) ) ).

% subset_mset.cInf_insert
tff(fact_7129_subset__mset_OcInf__le__cSup,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
       => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5)
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5))) ) ) ) ).

% subset_mset.cInf_le_cSup
tff(fact_7130_subset__mset_Oless__eq__cInf__inter,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5)
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B4)
       => ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4) != bot_bot(set(multiset(A))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B4))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4)))) ) ) ) ).

% subset_mset.less_eq_cInf_inter
tff(fact_7131_subset__mset_OcINF__inf__distrib,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A5))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A5))) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_amf(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A5)) ) ) ) ) ).

% subset_mset.cINF_inf_distrib
tff(fact_7132_subset__mset_OcInf__union__distrib,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A5)
       => ( ( B4 != bot_bot(set(multiset(A))) )
         => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B4)
           => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A5),B4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A5)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B4)) ) ) ) ) ) ).

% subset_mset.cInf_union_distrib
tff(fact_7133_subset__mset_OcInf__cSup,axiom,
    ! [A: $tType,S: set(multiset(A))] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),S)
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aTP_Lamp_amg(set(multiset(A)),fun(multiset(A),bool),S))) ) ) ) ).

% subset_mset.cInf_cSup
tff(fact_7134_subset__mset_OcINF__insert,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),A3: B] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A5))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(B,multiset(A),F3,A3)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))) ) ) ) ).

% subset_mset.cINF_insert
tff(fact_7135_subset__mset_OcINF__union,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),B4: set(B)] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( ( B4 != bot_bot(set(B)) )
         => ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B4))
           => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B4))) ) ) ) ) ) ).

% subset_mset.cINF_union
tff(fact_7136_repeat__mset__def,axiom,
    ! [A: $tType] : repeat_mset(A) = aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(nat,fun(multiset(A),multiset(A))),map_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(nat),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_amh(nat,fun(fun(A,nat),fun(A,nat)))) ).

% repeat_mset_def
tff(fact_7137_fold__mset__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),S2: B,M6: multiset(A)] : fold_mset(A,B,F3,S2,M6) = finite_fold(A,B,aa(multiset(A),fun(A,fun(B,B)),aTP_Lamp_ami(fun(A,fun(B,B)),fun(multiset(A),fun(A,fun(B,B))),F3),M6),S2,aa(multiset(A),set(A),set_mset(A),M6)) ).

% fold_mset_def
tff(fact_7138_mset__subseteq__add__iff1,axiom,
    ! [A: $tType,J: nat,I2: nat,U: multiset(A),M: multiset(A),N2: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J),U)),N2)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),M)),N2)) ) ) ).

% mset_subseteq_add_iff1
tff(fact_7139_mset__subseteq__add__iff2,axiom,
    ! [A: $tType,I2: nat,J: nat,U: multiset(A),M: multiset(A),N2: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J),U)),N2)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),N2))) ) ) ).

% mset_subseteq_add_iff2
tff(fact_7140_mset__subset__add__iff2,axiom,
    ! [A: $tType,I2: nat,J: nat,U: multiset(A),M: multiset(A),N2: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J),U)),N2)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),N2))) ) ) ).

% mset_subset_add_iff2
tff(fact_7141_mset__subset__add__iff1,axiom,
    ! [A: $tType,J: nat,I2: nat,U: multiset(A),M: multiset(A),N2: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J),U)),N2)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),M)),N2)) ) ) ).

% mset_subset_add_iff1
tff(fact_7142_sorted__list__of__multiset__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: multiset(A)] : linord6283353356039996273ltiset(A,M6) = fold_mset(A,list(A),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),nil(A),M6) ) ).

% sorted_list_of_multiset_def
tff(fact_7143_subset__mset_OInf__fin_Oremove,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( pp(aa(set(multiset(A)),bool,member(multiset(A),X),A5))
       => ( ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5) = X ) )
          & ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) != bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))))) ) ) ) ) ) ).

% subset_mset.Inf_fin.remove
tff(fact_7144_subset__mset_OInf__fin_Oinsert__remove,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = X ) )
        & ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) != bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))))) ) ) ) ) ).

% subset_mset.Inf_fin.insert_remove
tff(fact_7145_subset__mset_OInf__fin_Osingleton,axiom,
    ! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.Inf_fin.singleton
tff(fact_7146_subset__mset_OInf__fin_Oinsert,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)) ) ) ) ).

% subset_mset.Inf_fin.insert
tff(fact_7147_subset__mset_OInf__fin_Oeq__fold_H,axiom,
    ! [A: $tType,A5: set(multiset(A))] : aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5) = aa(option(multiset(A)),multiset(A),the2(multiset(A)),finite_fold(multiset(A),option(multiset(A)),aTP_Lamp_amj(multiset(A),fun(option(multiset(A)),option(multiset(A)))),none(multiset(A)),A5)) ).

% subset_mset.Inf_fin.eq_fold'
tff(fact_7148_semilattice__inf_OInf__fin_Ocong,axiom,
    ! [A: $tType,Inf2: fun(A,fun(A,A))] : lattic8678736583308907530nf_fin(A,Inf2) = lattic8678736583308907530nf_fin(A,Inf2) ).

% semilattice_inf.Inf_fin.cong
tff(fact_7149_subset__mset_OInf__fin_Ohom__commute,axiom,
    ! [A: $tType,H: fun(multiset(A),multiset(A)),N7: set(multiset(A))] :
      ( ! [X3: multiset(A),Y3: multiset(A)] : aa(multiset(A),multiset(A),H,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X3),Y3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(multiset(A),multiset(A),H,X3)),aa(multiset(A),multiset(A),H,Y3))
     => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),N7))
       => ( ( N7 != bot_bot(set(multiset(A))) )
         => ( aa(multiset(A),multiset(A),H,aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),N7)) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),H),N7)) ) ) ) ) ).

% subset_mset.Inf_fin.hom_commute
tff(fact_7150_subset__mset_OInf__fin_Obounded__iff,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)))
        <=> ! [X4: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),X4)) ) ) ) ) ).

% subset_mset.Inf_fin.bounded_iff
tff(fact_7151_subset__mset_OInf__fin_OboundedI,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( ! [A6: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),A6),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),A6)) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5))) ) ) ) ).

% subset_mset.Inf_fin.boundedI
tff(fact_7152_subset__mset_OInf__fin_OboundedE,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)))
         => ! [A14: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),A14),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),A14)) ) ) ) ) ).

% subset_mset.Inf_fin.boundedE
tff(fact_7153_subset__mset_OcInf__eq__Inf__fin,axiom,
    ! [A: $tType,X6: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),X6))
     => ( ( X6 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X6) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),X6) ) ) ) ).

% subset_mset.cInf_eq_Inf_fin
tff(fact_7154_subset__mset_OInf__fin_Oclosed,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( ! [X3: multiset(A),Y3: multiset(A)] : pp(aa(set(multiset(A)),bool,member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X3),Y3)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X3),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),Y3),bot_bot(set(multiset(A)))))))
         => pp(aa(set(multiset(A)),bool,member(multiset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)),A5)) ) ) ) ).

% subset_mset.Inf_fin.closed
tff(fact_7155_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ~ pp(aa(set(multiset(A)),bool,member(multiset(A),X),A5))
       => ( ( A5 != bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)) ) ) ) ) ).

% subset_mset.Inf_fin.insert_not_elem
tff(fact_7156_subset__mset_OInf__fin_Osubset,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( B4 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),B4),A5))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B4)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5) ) ) ) ) ).

% subset_mset.Inf_fin.subset
tff(fact_7157_subset__mset_OInf__fin_Ounion,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B4))
         => ( ( B4 != bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A5),B4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B4)) ) ) ) ) ) ).

% subset_mset.Inf_fin.union
tff(fact_7158_subset__mset_OInf__fin_Osubset__imp,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A5),B4))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B4))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B4)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5))) ) ) ) ).

% subset_mset.Inf_fin.subset_imp
tff(fact_7159_repeat__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa2: nat] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),X),X))
     => ( aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),Xa2),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(nat,fun(A,nat),aTP_Lamp_amk(fun(A,nat),fun(nat,fun(A,nat)),X),Xa2)) ) ) ).

% repeat_mset.abs_eq
tff(fact_7160_subset__mset_OcSUP__union,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),B4: set(B)] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( ( B4 != bot_bot(set(B)) )
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B4))
           => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B4))) ) ) ) ) ) ).

% subset_mset.cSUP_union
tff(fact_7161_set__mset__sup,axiom,
    ! [A: $tType,A5: multiset(A),B4: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A5),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(multiset(A),set(A),set_mset(A),A5)),aa(multiset(A),set(A),set_mset(A),B4)) ).

% set_mset_sup
tff(fact_7162_subset__mset_Obdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),G3: fun(B,multiset(A)),A5: set(B)] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_aml(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A5))
    <=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
        & condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A5)) ) ) ).

% subset_mset.bdd_above_image_sup
tff(fact_7163_eq__onp__mono__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),bNF_eq_onp(A,P)),bNF_eq_onp(A,Q)))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q)) ) ).

% eq_onp_mono_iff
tff(fact_7164_eq__onp__le__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] : pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),bNF_eq_onp(A,P)),fequal(A))) ).

% eq_onp_le_eq
tff(fact_7165_rel__fun__eq__onp__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: fun(A,bool),S: fun(B,fun(C,bool)),X5: fun(A,B),Xa3: fun(A,C)] :
      ( pp(aa(fun(A,C),bool,aa(fun(A,B),fun(fun(A,C),bool),bNF_rel_fun(A,A,B,C,bNF_eq_onp(A,R2),S),X5),Xa3))
    <=> ! [Xb5: A] :
          ( pp(aa(A,bool,R2,Xb5))
         => pp(aa(C,bool,aa(B,fun(C,bool),S,aa(A,B,X5,Xb5)),aa(A,C,Xa3,Xb5))) ) ) ).

% rel_fun_eq_onp_rel
tff(fact_7166_rel__fun__eq__eq__onp,axiom,
    ! [A: $tType,B: $tType,P: fun(B,bool)] : bNF_rel_fun(A,A,B,B,fequal(A),bNF_eq_onp(B,P)) = bNF_eq_onp(fun(A,B),aTP_Lamp_amm(fun(B,bool),fun(fun(A,B),bool),P)) ).

% rel_fun_eq_eq_onp
tff(fact_7167_eq__onp__True,axiom,
    ! [A: $tType] : bNF_eq_onp(A,aTP_Lamp_bm(A,bool)) = fequal(A) ).

% eq_onp_True
tff(fact_7168_eq__onp__def,axiom,
    ! [A: $tType,R2: fun(A,bool),X5: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,R2),X5),Xa3))
    <=> ( pp(aa(A,bool,R2,X5))
        & ( X5 = Xa3 ) ) ) ).

% eq_onp_def
tff(fact_7169_subset__mset_OSup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] : lattic4895041142388067077er_set(multiset(A),union_mset(A),aTP_Lamp_acu(multiset(A),fun(multiset(A),bool)),aTP_Lamp_acv(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.Sup_fin.semilattice_order_set_axioms
tff(fact_7170_zero__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),aTP_Lamp_amb(A,nat)),aTP_Lamp_amb(A,nat))) ).

% zero_multiset.rsp
tff(fact_7171_filter__mset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),bool,aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),bool),bNF_rel_fun(fun(A,bool),fun(A,bool),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(fun(A,bool)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)))),aTP_Lamp_amn(fun(A,bool),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_amn(fun(A,bool),fun(fun(A,nat),fun(A,nat))))) ).

% filter_mset.rsp
tff(fact_7172_subset__mset_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] : semila1105856199041335345_order(multiset(A),union_mset(A),zero_zero(multiset(A)),aTP_Lamp_acu(multiset(A),fun(multiset(A),bool)),aTP_Lamp_acv(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.semilattice_neutr_order_axioms
tff(fact_7173_add__mset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(A,fun(fun(A,nat),fun(A,nat))),bool,aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(fun(A,fun(fun(A,nat),fun(A,nat))),bool),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_alo(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)))),aTP_Lamp_ame(A,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_ame(A,fun(fun(A,nat),fun(A,nat))))) ).

% add_mset.rsp
tff(fact_7174_plus__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool),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_alo(fun(A,nat),bool)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)))),aTP_Lamp_amc(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_amc(fun(A,nat),fun(fun(A,nat),fun(A,nat))))) ).

% plus_multiset.rsp
tff(fact_7175_minus__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool),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_alo(fun(A,nat),bool)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)))),aTP_Lamp_amd(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_amd(fun(A,nat),fun(fun(A,nat),fun(A,nat))))) ).

% minus_multiset.rsp
tff(fact_7176_repeat__mset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(nat,fun(fun(A,nat),fun(A,nat))),bool,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(fun(A,nat),fun(A,nat))),bool),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_alo(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)))),aTP_Lamp_amh(nat,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_amh(nat,fun(fun(A,nat),fun(A,nat))))) ).

% repeat_mset.rsp
tff(fact_7177_subset__mset_Omono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F3: fun(multiset(A),B),A5: multiset(A),B4: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(multiset(A),B,F3,A5)),aa(multiset(A),B,F3,B4))),aa(multiset(A),B,F3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A5),B4)))) ) ) ).

% subset_mset.mono_sup
tff(fact_7178_subset__mset_OcSup__insert__If,axiom,
    ! [A: $tType,X6: set(multiset(A)),A3: multiset(A)] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X6)
     => ( ( ( X6 = bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),X6)) = A3 ) )
        & ( ( X6 != bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),X6)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X6)) ) ) ) ) ).

% subset_mset.cSup_insert_If
tff(fact_7179_subset__mset_OcSup__insert,axiom,
    ! [A: $tType,X6: set(multiset(A)),A3: multiset(A)] :
      ( ( X6 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X6)
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),A3),X6)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X6)) ) ) ) ).

% subset_mset.cSup_insert
tff(fact_7180_subset__mset_OcSup__inter__less__eq,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B4)
       => ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4) != bot_bot(set(multiset(A))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A5),B4))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B4)))) ) ) ) ).

% subset_mset.cSup_inter_less_eq
tff(fact_7181_subset__mset_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A5))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A5))) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_aml(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A5)) ) ) ) ) ).

% subset_mset.SUP_sup_distrib
tff(fact_7182_subset__mset_OcSup__union__distrib,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( ( A5 != bot_bot(set(multiset(A))) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A5)
       => ( ( B4 != bot_bot(set(multiset(A))) )
         => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B4)
           => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A5),B4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A5)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B4)) ) ) ) ) ) ).

% subset_mset.cSup_union_distrib
tff(fact_7183_subset__mset_Osup__Inf2__distrib,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B4))
         => ( ( B4 != bot_bot(set(multiset(A))) )
           => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B4)) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_amo(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),A5),B4))) ) ) ) ) ) ).

% subset_mset.sup_Inf2_distrib
tff(fact_7184_subset__mset_Osup__Inf1__distrib,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_amp(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),A5),X))) ) ) ) ).

% subset_mset.sup_Inf1_distrib
tff(fact_7185_add__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa2: A] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),X),X))
     => ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Xa2),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(A,fun(A,nat),aTP_Lamp_amq(fun(A,nat),fun(A,fun(A,nat)),X),Xa2)) ) ) ).

% add_mset.abs_eq
tff(fact_7186_plus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa2: fun(A,nat),X: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),Xa2),Xa2))
     => ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),X),X))
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(fun(A,nat),multiset(A),abs_multiset(A),Xa2)),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_amc(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa2),X)) ) ) ) ).

% plus_multiset.abs_eq
tff(fact_7187_minus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa2: fun(A,nat),X: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),Xa2),Xa2))
     => ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),X),X))
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(fun(A,nat),multiset(A),abs_multiset(A),Xa2)),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_amd(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa2),X)) ) ) ) ).

% minus_multiset.abs_eq
tff(fact_7188_subset__mset_OcSUP__insert,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F3: fun(B,multiset(A)),A3: B] :
      ( ( A5 != bot_bot(set(B)) )
     => ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A5))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(B,multiset(A),F3,A3)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A5))) ) ) ) ).

% subset_mset.cSUP_insert
tff(fact_7189_subset__mset_OSup__fin_Oeq__fold_H,axiom,
    ! [A: $tType,A5: set(multiset(A))] : aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5) = aa(option(multiset(A)),multiset(A),the2(multiset(A)),finite_fold(multiset(A),option(multiset(A)),aTP_Lamp_amr(multiset(A),fun(option(multiset(A)),option(multiset(A)))),none(multiset(A)),A5)) ).

% subset_mset.Sup_fin.eq_fold'
tff(fact_7190_Inf__multiset_Oabs__eq,axiom,
    ! [A: $tType,X: set(fun(A,nat))] :
      ( pp(aa(set(fun(A,nat)),bool,aa(set(fun(A,nat)),fun(set(fun(A,nat)),bool),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool))),X),X))
     => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(fun(A,nat)),set(multiset(A)),image2(fun(A,nat),multiset(A),abs_multiset(A)),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_alz(set(fun(A,nat)),fun(A,nat)),X)) ) ) ).

% Inf_multiset.abs_eq
tff(fact_7191_subset__mset_OSup__fin_Osingleton,axiom,
    ! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).

% subset_mset.Sup_fin.singleton
tff(fact_7192_subset__mset_OSup__fin_Oinsert,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)) ) ) ) ).

% subset_mset.Sup_fin.insert
tff(fact_7193_semilattice__sup_OSup__fin_Ocong,axiom,
    ! [A: $tType,Sup2: fun(A,fun(A,A))] : lattic4630905495605216202up_fin(A,Sup2) = lattic4630905495605216202up_fin(A,Sup2) ).

% semilattice_sup.Sup_fin.cong
tff(fact_7194_rel__set__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X5: set(A),Xa3: set(B)] :
      ( pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,R2),X5),Xa3))
    <=> ( ! [Xb5: A] :
            ( pp(aa(set(A),bool,member(A,Xb5),X5))
           => ? [Xc2: B] :
                ( pp(aa(set(B),bool,member(B,Xc2),Xa3))
                & pp(aa(B,bool,aa(A,fun(B,bool),R2,Xb5),Xc2)) ) )
        & ! [Xb5: B] :
            ( pp(aa(set(B),bool,member(B,Xb5),Xa3))
           => ? [Xc2: A] :
                ( pp(aa(set(A),bool,member(A,Xc2),X5))
                & pp(aa(B,bool,aa(A,fun(B,bool),R2,Xc2),Xb5)) ) ) ) ) ).

% rel_set_def
tff(fact_7195_fun_Oset__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(fun(D,B),set(B)),bool,aa(fun(fun(D,A),set(A)),fun(fun(fun(D,B),set(B)),bool),bNF_rel_fun(fun(D,A),fun(D,B),set(A),set(B),bNF_rel_fun(D,D,A,B,fequal(D),R2),bNF_rel_set(A,B,R2)),aTP_Lamp_ams(fun(D,A),set(A))),aTP_Lamp_amt(fun(D,B),set(B)))) ).

% fun.set_transfer
tff(fact_7196_subset__mset_OSup__fin_OboundedE,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)),X))
         => ! [A14: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),A14),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A14),X)) ) ) ) ) ).

% subset_mset.Sup_fin.boundedE
tff(fact_7197_subset__mset_OSup__fin_OboundedI,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( ! [A6: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),A6),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),X)) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)),X)) ) ) ) ).

% subset_mset.Sup_fin.boundedI
tff(fact_7198_subset__mset_OSup__fin_Obounded__iff,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)),X))
        <=> ! [X4: multiset(A)] :
              ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),A5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X4),X)) ) ) ) ) ).

% subset_mset.Sup_fin.bounded_iff
tff(fact_7199_subset__mset_OcSup__eq__Sup__fin,axiom,
    ! [A: $tType,X6: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),X6))
     => ( ( X6 != bot_bot(set(multiset(A))) )
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X6) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),X6) ) ) ) ).

% subset_mset.cSup_eq_Sup_fin
tff(fact_7200_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ~ pp(aa(set(multiset(A)),bool,member(multiset(A),X),A5))
       => ( ( A5 != bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)) ) ) ) ) ).

% subset_mset.Sup_fin.insert_not_elem
tff(fact_7201_subset__mset_OSup__fin_Oclosed,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( ! [X3: multiset(A),Y3: multiset(A)] : pp(aa(set(multiset(A)),bool,member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X3),Y3)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X3),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),Y3),bot_bot(set(multiset(A)))))))
         => pp(aa(set(multiset(A)),bool,member(multiset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)),A5)) ) ) ) ).

% subset_mset.Sup_fin.closed
tff(fact_7202_subset__mset_OSup__fin_Osubset,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( B4 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),B4),A5))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B4)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5) ) ) ) ) ).

% subset_mset.Sup_fin.subset
tff(fact_7203_subset__mset_OSup__fin_Ounion,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B4))
         => ( ( B4 != bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A5),B4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B4)) ) ) ) ) ) ).

% subset_mset.Sup_fin.union
tff(fact_7204_subset__mset_OSup__fin_Ohom__commute,axiom,
    ! [A: $tType,H: fun(multiset(A),multiset(A)),N7: set(multiset(A))] :
      ( ! [X3: multiset(A),Y3: multiset(A)] : aa(multiset(A),multiset(A),H,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X3),Y3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(multiset(A),multiset(A),H,X3)),aa(multiset(A),multiset(A),H,Y3))
     => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),N7))
       => ( ( N7 != bot_bot(set(multiset(A))) )
         => ( aa(multiset(A),multiset(A),H,aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),N7)) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),H),N7)) ) ) ) ) ).

% subset_mset.Sup_fin.hom_commute
tff(fact_7205_subset__mset_OSup__fin_Osubset__imp,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A5),B4))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B4))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B4))) ) ) ) ).

% subset_mset.Sup_fin.subset_imp
tff(fact_7206_subset__mset_Oinf__Sup1__distrib,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_amu(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),A5),X))) ) ) ) ).

% subset_mset.inf_Sup1_distrib
tff(fact_7207_subset__mset_Oinf__Sup2__distrib,axiom,
    ! [A: $tType,A5: set(multiset(A)),B4: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B4))
         => ( ( B4 != bot_bot(set(multiset(A))) )
           => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B4)) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_amv(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),A5),B4))) ) ) ) ) ) ).

% subset_mset.inf_Sup2_distrib
tff(fact_7208_subset__mset_OSup__fin_Oremove,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( pp(aa(set(multiset(A)),bool,member(multiset(A),X),A5))
       => ( ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5) = X ) )
          & ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) != bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))))) ) ) ) ) ) ).

% subset_mset.Sup_fin.remove
tff(fact_7209_subset__mset_OSup__fin_Oinsert__remove,axiom,
    ! [A: $tType,A5: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = X ) )
        & ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))) != bot_bot(set(multiset(A))) )
         => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),A5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),A5),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert(multiset(A)),X),bot_bot(set(multiset(A))))))) ) ) ) ) ).

% subset_mset.Sup_fin.insert_remove
tff(fact_7210_subset__mset_OInf__fin__le__Sup__fin,axiom,
    ! [A: $tType,A5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A5))
     => ( ( A5 != bot_bot(set(multiset(A))) )
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A5)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A5))) ) ) ).

% subset_mset.Inf_fin_le_Sup_fin
tff(fact_7211_Inf__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(set(fun(A,nat)),fun(A,nat)),bool,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(fun(A,nat)),fun(A,nat)),bool),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_alo(fun(A,nat),bool))),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool))),aTP_Lamp_alz(set(fun(A,nat)),fun(A,nat))),aTP_Lamp_alz(set(fun(A,nat)),fun(A,nat)))) ).

% Inf_multiset.rsp
tff(fact_7212_INF__parametric,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [A5: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(fun(B,C),C)),bool,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,A5),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,A5,fequal(C)),fequal(C))),aTP_Lamp_amw(set(A),fun(fun(A,C),C))),aTP_Lamp_amx(set(B),fun(fun(B,C),C)))) ) ).

% INF_parametric
tff(fact_7213_SUP__parametric,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [R2: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(fun(B,C),C)),bool,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,R2),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,R2,fequal(C)),fequal(C))),aTP_Lamp_amy(set(A),fun(fun(A,C),C))),aTP_Lamp_amz(set(B),fun(fun(B,C),C)))) ) ).

% SUP_parametric
tff(fact_7214_empty__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] : pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,A5),bot_bot(set(A))),bot_bot(set(B)))) ).

% empty_transfer
tff(fact_7215_union__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(set(B),set(B))),bool,aa(fun(set(A),fun(set(A),set(A))),fun(fun(set(B),fun(set(B),set(B))),bool),bNF_rel_fun(set(A),set(B),fun(set(A),set(A)),fun(set(B),set(B)),bNF_rel_set(A,B,A5),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A5),bNF_rel_set(A,B,A5))),sup_sup(set(A))),sup_sup(set(B)))) ).

% union_transfer
tff(fact_7216_rel__set__mono,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,fun(B,bool)),B4: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A5),B4))
     => pp(aa(fun(set(A),fun(set(B),bool)),bool,aa(fun(set(A),fun(set(B),bool)),fun(fun(set(A),fun(set(B),bool)),bool),ord_less_eq(fun(set(A),fun(set(B),bool))),bNF_rel_set(A,B,A5)),bNF_rel_set(A,B,B4))) ) ).

% rel_set_mono
tff(fact_7217_Union__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] : pp(aa(fun(set(set(B)),set(B)),bool,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),bool),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A5)),bNF_rel_set(A,B,A5)),complete_Sup_Sup(set(A))),complete_Sup_Sup(set(B)))) ).

% Union_transfer
tff(fact_7218_set__relator__eq__onp,axiom,
    ! [A: $tType,P: fun(A,bool)] : bNF_rel_set(A,A,bNF_eq_onp(A,P)) = bNF_eq_onp(set(A),aTP_Lamp_ana(fun(A,bool),fun(set(A),bool),P)) ).

% set_relator_eq_onp
tff(fact_7219_UNION__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A5: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool))] : pp(aa(fun(set(B),fun(fun(B,set(D)),set(D))),bool,aa(fun(set(A),fun(fun(A,set(C)),set(C))),fun(fun(set(B),fun(fun(B,set(D)),set(D))),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,set(C)),set(C)),fun(fun(B,set(D)),set(D)),bNF_rel_set(A,B,A5),bNF_rel_fun(fun(A,set(C)),fun(B,set(D)),set(C),set(D),bNF_rel_fun(A,B,set(C),set(D),A5,bNF_rel_set(C,D,B4)),bNF_rel_set(C,D,B4))),aTP_Lamp_anb(set(A),fun(fun(A,set(C)),set(C)))),aTP_Lamp_anc(set(B),fun(fun(B,set(D)),set(D))))) ).

% UNION_transfer
tff(fact_7220_Inf__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(set(multiset(A)),multiset(A)),bool,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(multiset(A)),multiset(A)),bool),bNF_rel_fun(set(fun(A,nat)),set(multiset(A)),fun(A,nat),multiset(A),bNF_rel_set(fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A))),pcr_multiset(A,A,fequal(A))),aTP_Lamp_alz(set(fun(A,nat)),fun(A,nat))),complete_Inf_Inf(multiset(A)))) ).

% Inf_multiset.transfer
tff(fact_7221_filter__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa2: fun(A,bool)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_alo(fun(A,nat),bool)),X),X))
     => ( aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),Xa2),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,bool),fun(A,nat),aTP_Lamp_and(fun(A,nat),fun(fun(A,bool),fun(A,nat)),X),Xa2)) ) ) ).

% filter_mset.abs_eq
tff(fact_7222_filter__mset__True,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_bm(A,bool)),M6) = M6 ).

% filter_mset_True
tff(fact_7223_filter__mset__False,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_ae(A,bool)),M6) = zero_zero(multiset(A)) ).

% filter_mset_False
tff(fact_7224_set__mset__filter,axiom,
    ! [A: $tType,P: fun(A,bool),M6: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P),M6)) = aa(fun(A,bool),set(A),collect(A),aa(multiset(A),fun(A,bool),aTP_Lamp_ane(fun(A,bool),fun(multiset(A),fun(A,bool)),P),M6)) ).

% set_mset_filter
tff(fact_7225_mset__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : mset(A,aa(list(A),list(A),filter2(A,P),Xs)) = aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P),mset(A,Xs)) ).

% mset_filter
tff(fact_7226_filter__mset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,bool),fun(multiset(A),multiset(A))),bool,aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,bool),fun(multiset(A),multiset(A))),bool),bNF_rel_fun(fun(A,bool),fun(A,bool),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(fun(A,bool)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_amn(fun(A,bool),fun(fun(A,nat),fun(A,nat)))),filter_mset(A))) ).

% filter_mset.transfer
tff(fact_7227_filter__filter__mset,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),M6: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),Q),M6)) = aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_sl(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)),M6) ).

% filter_filter_mset
tff(fact_7228_multiset__partition,axiom,
    ! [A: $tType,M6: multiset(A),P: fun(A,bool)] : M6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P),M6)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_ci(fun(A,bool),fun(A,bool),P)),M6)) ).

% multiset_partition
tff(fact_7229_image__mset__If,axiom,
    ! [A: $tType,B: $tType,P: fun(B,bool),F3: fun(B,A),G3: fun(B,A),A5: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ct(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F3),G3)),A5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(B),multiset(A),image_mset(B,A,F3),aa(multiset(B),multiset(B),aa(fun(B,bool),fun(multiset(B),multiset(B)),filter_mset(B),P),A5))),aa(multiset(B),multiset(A),image_mset(B,A,G3),aa(multiset(B),multiset(B),aa(fun(B,bool),fun(multiset(B),multiset(B)),filter_mset(B),aTP_Lamp_cu(fun(B,bool),fun(B,bool),P)),A5))) ).

% image_mset_If
tff(fact_7230_size__filter__mset__lesseq,axiom,
    ! [A: $tType,F3: fun(A,bool),M6: multiset(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),F3),M6))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ).

% size_filter_mset_lesseq
tff(fact_7231_filter__eq__replicate__mset,axiom,
    ! [A: $tType,X: A,D4: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_br(A,fun(A,bool),X)),D4) = replicate_mset(A,aa(A,nat,aa(multiset(A),fun(A,nat),count(A),D4),X),X) ).

% filter_eq_replicate_mset
tff(fact_7232_zero__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(multiset(A),bool,aa(fun(A,nat),fun(multiset(A),bool),pcr_multiset(A,A,fequal(A)),aTP_Lamp_amb(A,nat)),zero_zero(multiset(A)))) ).

% zero_multiset.transfer
tff(fact_7233_multiset_Orep__transfer,axiom,
    ! [D: $tType,E: $tType,T2: fun(D,fun(E,bool))] : pp(aa(fun(multiset(E),fun(E,nat)),bool,aa(fun(fun(D,nat),fun(D,nat)),fun(fun(multiset(E),fun(E,nat)),bool),bNF_rel_fun(fun(D,nat),multiset(E),fun(D,nat),fun(E,nat),pcr_multiset(D,E,T2),bNF_rel_fun(D,E,nat,nat,T2,fequal(nat))),aTP_Lamp_anf(fun(D,nat),fun(D,nat))),count(E))) ).

% multiset.rep_transfer
tff(fact_7234_add__mset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(A,fun(multiset(A),multiset(A))),bool,aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(fun(A,fun(multiset(A),multiset(A))),bool),bNF_rel_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(A),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_ame(A,fun(fun(A,nat),fun(A,nat)))),add_mset(A))) ).

% add_mset.transfer
tff(fact_7235_plus__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(multiset(A),fun(multiset(A),multiset(A))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(multiset(A),fun(multiset(A),multiset(A))),bool),bNF_rel_fun(fun(A,nat),multiset(A),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),pcr_multiset(A,A,fequal(A)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_amc(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),plus_plus(multiset(A)))) ).

% plus_multiset.transfer
tff(fact_7236_minus__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(multiset(A),fun(multiset(A),multiset(A))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(multiset(A),fun(multiset(A),multiset(A))),bool),bNF_rel_fun(fun(A,nat),multiset(A),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),pcr_multiset(A,A,fequal(A)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_amd(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),minus_minus(multiset(A)))) ).

% minus_multiset.transfer
tff(fact_7237_repeat__mset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(nat,fun(multiset(A),multiset(A))),bool,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(multiset(A),multiset(A))),bool),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(nat),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_amh(nat,fun(fun(A,nat),fun(A,nat)))),repeat_mset(A))) ).

% repeat_mset.transfer
tff(fact_7238_filter__mset__def,axiom,
    ! [A: $tType] : filter_mset(A) = aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),fun(fun(A,bool),fun(multiset(A),multiset(A))),map_fun(fun(A,bool),fun(A,bool),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(fun(A,bool)),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_amn(fun(A,bool),fun(fun(A,nat),fun(A,nat)))) ).

% filter_mset_def
tff(fact_7239_type__definition__multiset,axiom,
    ! [A: $tType] : type_definition(multiset(A),fun(A,nat),count(A),abs_multiset(A),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_alo(fun(A,nat),bool))) ).

% type_definition_multiset
tff(fact_7240_sum__multiset__singleton,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),multiset(A),aa(fun(A,multiset(A)),fun(set(A),multiset(A)),groups7311177749621191930dd_sum(A,multiset(A)),aTP_Lamp_akb(A,multiset(A))),A5) = mset_set(A,A5) ).

% sum_multiset_singleton
tff(fact_7241_mset__set_Oempty,axiom,
    ! [A: $tType] : mset_set(A,bot_bot(set(A))) = zero_zero(multiset(A)) ).

% mset_set.empty
tff(fact_7242_filter__mset__mset__set,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P),mset_set(A,A5)) = mset_set(A,aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),A5),P))) ) ) ).

% filter_mset_mset_set
tff(fact_7243_msubset__mset__set__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),mset_set(A,A5)),mset_set(A,B4)))
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4)) ) ) ) ).

% msubset_mset_set_iff
tff(fact_7244_typedef__rep__transfer,axiom,
    ! [A: $tType,B: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T2: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T2,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => pp(aa(fun(B,A),bool,aa(fun(A,A),fun(fun(B,A),bool),bNF_rel_fun(A,B,A,A,T2,fequal(A)),aTP_Lamp_cq(A,A)),Rep)) ) ) ).

% typedef_rep_transfer
tff(fact_7245_subset__imp__msubset__mset__set,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),mset_set(A,A5)),mset_set(A,B4))) ) ) ).

% subset_imp_msubset_mset_set
tff(fact_7246_mset__set__empty__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( mset_set(A,A5) = zero_zero(multiset(A)) )
    <=> ( ( A5 = bot_bot(set(A)) )
        | ~ pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% mset_set_empty_iff
tff(fact_7247_infinite__set__mset__mset__set,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(multiset(A),set(A),set_mset(A),mset_set(A,A5)) = bot_bot(set(A)) ) ) ).

% infinite_set_mset_mset_set
tff(fact_7248_mset__set__Diff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),mset_set(A,A5)),mset_set(A,B4)) ) ) ) ).

% mset_set_Diff
tff(fact_7249_count__mset__set__finite__iff,axiom,
    ! [A: $tType,S: set(A),A3: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( ( pp(aa(set(A),bool,member(A,A3),S))
         => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S)),A3) = one_one(nat) ) )
        & ( ~ pp(aa(set(A),bool,member(A,A3),S))
         => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S)),A3) = zero_zero(nat) ) ) ) ) ).

% count_mset_set_finite_iff
tff(fact_7250_mset__set_Oremove,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( mset_set(A,A5) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% mset_set.remove
tff(fact_7251_mset__set_Oinsert__remove,axiom,
    ! [A: $tType,A5: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( mset_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A5)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ).

% mset_set.insert_remove
tff(fact_7252_mset__set__Union,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
         => ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,A5)),mset_set(A,B4)) ) ) ) ) ).

% mset_set_Union
tff(fact_7253_stable__sort__key__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Sk: fun(fun(B,A),fun(list(B),list(B)))] :
          ( linord3483353639454293061rt_key(B,A,Sk)
        <=> ! [F10: fun(B,A),Xs3: list(B),K3: A] : aa(list(B),list(B),filter2(B,aa(A,fun(B,bool),aTP_Lamp_tx(fun(B,A),fun(A,fun(B,bool)),F10),K3)),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),Sk,F10),Xs3)) = aa(list(B),list(B),filter2(B,aa(A,fun(B,bool),aTP_Lamp_tx(fun(B,A),fun(A,fun(B,bool)),F10),K3)),Xs3) ) ) ).

% stable_sort_key_def
tff(fact_7254_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F4: fun(A,set(B)),F3: fun(B,C),G4: fun(D,set(C)),J5: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => ! [J2: A] :
                ( pp(aa(set(A),bool,member(A,J2),I5))
               => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F4,I3)),aa(A,set(B),F4,J2)))
                  | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F4,J2)),aa(A,set(B),F4,I3))) ) ) )
       => ( filterlim(B,C,F3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_ang(fun(D,set(C)),fun(D,filter(C)),G4)),J5)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_adl(fun(A,set(B)),fun(A,filter(B)),F4)),I5)))
        <=> ! [X4: D] :
              ( pp(aa(set(D),bool,member(D,X4),J5))
             => ? [Xa: A] :
                  ( pp(aa(set(A),bool,member(A,Xa),I5))
                  & ! [Xb5: B] :
                      ( pp(aa(set(B),bool,member(B,Xb5),aa(A,set(B),F4,Xa)))
                     => pp(aa(set(C),bool,member(C,aa(B,C,F3,Xb5)),aa(D,set(C),G4,X4))) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_7255_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: fun(A,B),F33: filter(B),F23: filter(A),F3: fun(C,A),F14: filter(C)] :
      ( filterlim(A,B,G3,F33,F23)
     => ( filterlim(C,A,F3,F23,F14)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_anh(fun(A,B),fun(fun(C,A),fun(C,B)),G3),F3),F33,F14) ) ) ).

% filterlim_compose
tff(fact_7256_filterlim__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filterlim(A,A,aTP_Lamp_cq(A,A),F4,F4) ).

% filterlim_ident
tff(fact_7257_filterlim__top,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(A)] : filterlim(A,B,F3,top_top(filter(B)),F4) ).

% filterlim_top
tff(fact_7258_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F23: filter(B),F33: filter(B),F14: filter(A)] :
      ( filterlim(A,B,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F33),F14)
    <=> ( filterlim(A,B,F3,F23,F14)
        & filterlim(A,B,F3,F33,F14) ) ) ).

% filterlim_inf
tff(fact_7259_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_7260_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_7261_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_7262_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_vy(fun(nat,A),fun(nat,A),F3),F4,at_top(nat))
    <=> filterlim(nat,A,F3,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_7263_filterlim__sup,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(B),F14: filter(A),F23: filter(A)] :
      ( filterlim(A,B,F3,F4,F14)
     => ( filterlim(A,B,F3,F4,F23)
       => filterlim(A,B,F3,F4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F14),F23)) ) ) ).

% filterlim_sup
tff(fact_7264_filterlim__INF,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,B),G4: fun(C,filter(B)),B4: set(C),F4: filter(A)] :
      ( filterlim(A,B,F3,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),G4),B4)),F4)
    <=> ! [X4: C] :
          ( pp(aa(set(C),bool,member(C,X4),B4))
         => filterlim(A,B,F3,aa(C,filter(B),G4,X4),F4) ) ) ).

% filterlim_INF
tff(fact_7265_filterlim__INF_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: A,A5: set(A),F3: fun(B,C),F4: filter(C),G4: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ( filterlim(B,C,F3,F4,aa(A,filter(B),G4,X))
       => filterlim(B,C,F3,F4,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),G4),A5))) ) ) ).

% filterlim_INF'
tff(fact_7266_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F23: filter(B),F14: filter(A),F24: filter(B),F15: filter(A)] :
      ( filterlim(A,B,F3,F23,F14)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F23),F24))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F15),F14))
         => filterlim(A,B,F3,F24,F15) ) ) ) ).

% filterlim_mono
tff(fact_7267_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G4: filter(B),F4: filter(A),P: fun(A,bool),G3: fun(A,B)] :
      ( filterlim(A,B,F3,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,aa(fun(A,bool),set(A),collect(A),P))))
     => ( filterlim(A,B,G3,G4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ci(fun(A,bool),fun(A,bool),P)))))
       => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_ani(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),F3),P),G3),G4,F4) ) ) ).

% filterlim_If
tff(fact_7268_filterlim__base,axiom,
    ! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J5: set(A),I2: fun(A,C),I5: set(C),F4: fun(C,set(D)),F3: fun(D,E),G4: fun(A,set(E))] :
      ( ! [M5: A,X3: B] :
          ( pp(aa(set(A),bool,member(A,M5),J5))
         => pp(aa(set(C),bool,member(C,aa(A,C,I2,M5)),I5)) )
     => ( ! [M5: A,X3: D] :
            ( pp(aa(set(A),bool,member(A,M5),J5))
           => ( pp(aa(set(D),bool,member(D,X3),aa(C,set(D),F4,aa(A,C,I2,M5))))
             => pp(aa(set(E),bool,member(E,aa(D,E,F3,X3)),aa(A,set(E),G4,M5))) ) )
       => filterlim(D,E,F3,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),aTP_Lamp_anj(fun(A,set(E)),fun(A,filter(E)),G4)),J5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),image2(C,filter(D),aTP_Lamp_ank(fun(C,set(D)),fun(C,filter(D)),F4)),I5))) ) ) ).

% filterlim_base
tff(fact_7269_map__rec,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),rec_list(list(A),B,nil(A),aTP_Lamp_anl(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F3)),Xs) ).

% map_rec
tff(fact_7270_zipf__zip,axiom,
    ! [A: $tType,B: $tType,L1: list(A),L22: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L22) )
     => ( zipf(A,B,product_prod(A,B),product_Pair(A,B),L1,L22) = zip(A,B,L1,L22) ) ) ).

% zipf_zip
tff(fact_7271_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_7272_rec__list__Cons__imp,axiom,
    ! [B: $tType,A: $tType,F3: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B))),X: A,Xs: list(A)] :
      ( ( F3 = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),F22,X),Xs),aa(list(A),B,F3,Xs)) ) ) ).

% rec_list_Cons_imp
tff(fact_7273_zipf_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,fun(B,C))] : zipf(A,B,C,F3,nil(A),nil(B)) = nil(C) ).

% zipf.simps(1)
tff(fact_7274_zipf_Osimps_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,fun(B,C)),A3: A,As: list(A),B2: B,Bs: list(B)] : zipf(A,B,C,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),F3,A3),B2)),zipf(A,B,C,F3,As,Bs)) ).

% zipf.simps(2)
tff(fact_7275_rec__list__Nil__imp,axiom,
    ! [A: $tType,B: $tType,F3: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B)))] :
      ( ( F3 = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F3,nil(A)) = F1 ) ) ).

% rec_list_Nil_imp
tff(fact_7276_zipf_Oelims,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: fun(A,fun(B,C)),Xa2: list(A),Xb: list(B),Y: list(C)] :
      ( ( zipf(A,B,C,X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = nil(A) )
         => ( ( Xb = nil(B) )
           => ( Y != nil(C) ) ) )
       => ( ! [A6: A,As4: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
             => ! [B5: B,Bs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
                 => ( Y != aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X,A6),B5)),zipf(A,B,C,X,As4,Bs2)) ) ) )
         => ( ( ? [V3: A,Va: list(A)] : Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)
             => ( ( Xb = nil(B) )
               => ( Y != undefined(list(C)) ) ) )
           => ~ ( ( Xa2 = 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 != undefined(list(C)) ) ) ) ) ) ) ) ).

% zipf.elims
tff(fact_7277_list_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: C,Ga: fun(B,fun(list(B),fun(C,C))),F3: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),C),comp(list(B),C,list(A),rec_list(C,B,G3,Ga)),map(A,B,F3)) = rec_list(C,A,G3,aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_anm(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Ga),F3)) ).

% list.rec_o_map
tff(fact_7278_zipf_Opelims,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: fun(A,fun(B,C)),Xa2: list(A),Xb: list(B),Y: list(C)] :
      ( ( zipf(A,B,C,X,Xa2,Xb) = Y )
     => ( pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa2),Xb))))
       => ( ( ( Xa2 = nil(A) )
           => ( ( Xb = nil(B) )
             => ( ( Y = nil(C) )
               => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))))) ) ) )
         => ( ! [A6: A,As4: list(A)] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
               => ! [B5: B,Bs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
                   => ( ( Y = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X,A6),B5)),zipf(A,B,C,X,As4,Bs2)) )
                     => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2))))) ) ) )
           => ( ! [V3: A,Va: list(A)] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( ( Xb = nil(B) )
                   => ( ( Y = undefined(list(C)) )
                     => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B))))) ) ) )
             => ~ ( ( Xa2 = 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 = undefined(list(C)) )
                       => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))))) ) ) ) ) ) ) ) ) ).

% zipf.pelims
tff(fact_7279_set__rec,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_ann(A,fun(list(A),fun(set(A),set(A))))),Xs) ).

% set_rec
tff(fact_7280_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ( semiring_gcd_Gcd_fin(A,A5) = zero_zero(A) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A)))))
            & pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_7281_map__tailrec__rev_Opelims,axiom,
    ! [A: $tType,B: $tType,X: fun(A,B),Xa2: list(A),Xb: list(B),Y: list(B)] :
      ( ( map_tailrec_rev(A,B,X,Xa2,Xb) = Y )
     => ( pp(aa(product_prod(fun(A,B),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B)),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))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa2),Xb))))
       => ( ( ( Xa2 = nil(A) )
           => ( ( Y = Xb )
             => ~ pp(aa(product_prod(fun(A,B),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B)),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))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xb)))) ) )
         => ~ ! [A6: A,As4: list(A)] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4) )
               => ( ( Y = map_tailrec_rev(A,B,X,As4,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,X,A6)),Xb)) )
                 => ~ pp(aa(product_prod(fun(A,B),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B)),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))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As4)),Xb)))) ) ) ) ) ) ).

% map_tailrec_rev.pelims
tff(fact_7282_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_7283_Gcd__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B4: set(A),A5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),semiring_gcd_Gcd_fin(A,B4)),semiring_gcd_Gcd_fin(A,A5)) = semiring_gcd_Gcd_fin(A,A5) ) ) ) ).

% Gcd_fin.subset
tff(fact_7284_Gcd__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B4: set(A)] : semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),semiring_gcd_Gcd_fin(A,A5)),semiring_gcd_Gcd_fin(A,B4)) ) ).

% Gcd_fin.union
tff(fact_7285_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] : semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A5)) = 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_7286_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( semiring_gcd_Gcd_fin(A,A5) = 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_7287_curr__in,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(product_prod(A,B),C),A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),F3),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),C4)))
     => pp(aa(set(fun(A,fun(B,C))),bool,member(fun(A,fun(B,C)),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),F3)),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C4)))) ) ).

% curr_in
tff(fact_7288_curr__surj,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: fun(A,fun(B,C)),A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(fun(A,fun(B,C))),bool,member(fun(A,fun(B,C)),G3),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C4))))
     => ? [X3: fun(product_prod(A,B),C)] :
          ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),X3),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),C4)))
          & ( aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),X3) = G3 ) ) ) ).

% curr_surj
tff(fact_7289_curr__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,A5: set(A),F3: fun(product_prod(A,B),C),X5: A] :
      ( ( pp(aa(set(A),bool,member(A,X5),A5))
       => ( aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),F3),X5) = aa(A,fun(B,C),aTP_Lamp_iw(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3),X5) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X5),A5))
       => ( aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),F3),X5) = undefined(fun(B,C)) ) ) ) ).

% curr_def
tff(fact_7290_curr__inj,axiom,
    ! [C: $tType,B: $tType,A: $tType,F1: fun(product_prod(A,B),C),A5: set(A),B4: set(B),C4: set(C),F22: fun(product_prod(A,B),C)] :
      ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),F1),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),C4)))
     => ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),F22),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),C4)))
       => ( ( aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),F1) = aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),F22) )
        <=> ( F1 = F22 ) ) ) ) ).

% curr_inj
tff(fact_7291_bij__betw__curr,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: set(A),B4: set(B),C4: set(C)] : bij_betw(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),C4),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C4))) ).

% bij_betw_curr
tff(fact_7292_disjnt__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A5,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
      <=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ).

% disjnt_equiv_class
tff(fact_7293_disjnt__self__iff__empty,axiom,
    ! [A: $tType,S: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),S),S))
    <=> ( S = bot_bot(set(A)) ) ) ).

% disjnt_self_iff_empty
tff(fact_7294_disjnt__insert2,axiom,
    ! [A: $tType,Y6: set(A),A3: A,X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)))
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),Y6))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y6),X6)) ) ) ).

% disjnt_insert2
tff(fact_7295_disjnt__insert1,axiom,
    ! [A: $tType,A3: A,X6: set(A),Y6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)),Y6))
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),Y6))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6)) ) ) ).

% disjnt_insert1
tff(fact_7296_disjnt__Un2,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),A5))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),B4)) ) ) ).

% disjnt_Un2
tff(fact_7297_disjnt__Un1,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)),C4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),C4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B4),C4)) ) ) ).

% disjnt_Un1
tff(fact_7298_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C4: set(A),A5: set(B),B4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,C4,aTP_Lamp_xt(set(B),fun(A,set(B)),A5))),product_Sigma(A,B,C4,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))
    <=> ( ( C4 = bot_bot(set(A)) )
        | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),A5),B4)) ) ) ).

% disjnt_Times1_iff
tff(fact_7299_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C4: set(B),B4: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B4,aTP_Lamp_xt(set(B),fun(A,set(B)),C4))))
    <=> ( ( C4 = bot_bot(set(B)) )
        | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),B4)) ) ) ).

% disjnt_Times2_iff
tff(fact_7300_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C4: fun(A,set(B)),B4: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A5,C4)),product_Sigma(A,B,B4,C4)))
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
           => ( aa(A,set(B),C4,X4) = bot_bot(set(B)) ) )
        | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),B4)) ) ) ).

% disjnt_Sigma_iff
tff(fact_7301_disjnt__def,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),B4))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) ) ) ).

% disjnt_def
tff(fact_7302_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( bij_betw(A,B,F3,A5,bot_bot(set(B)))
     => ( A5 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_7303_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] :
      ( bij_betw(A,B,F3,bot_bot(set(A)),A5)
     => ( A5 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_7304_disjnt__empty1,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),bot_bot(set(A))),A5)) ).

% disjnt_empty1
tff(fact_7305_disjnt__empty2,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),bot_bot(set(A)))) ).

% disjnt_empty2
tff(fact_7306_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A5: set(A),F3: fun(A,B),A15: set(B)] :
      ( ~ pp(aa(set(A),bool,member(A,B2),A5))
     => ( ~ pp(aa(set(B),bool,member(B,aa(A,B,F3,B2)),A15))
       => ( bij_betw(A,B,F3,A5,A15)
         => bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A15),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_7307_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A5: set(A),F3: fun(A,B),A15: set(B)] :
      ( ~ pp(aa(set(A),bool,member(A,B2),A5))
     => ( ~ pp(aa(set(B),bool,member(B,aa(A,B,F3,B2)),A15))
       => ( bij_betw(A,B,F3,A5,A15)
        <=> bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A15),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_7308_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),C4: set(A),B4: set(B),D4: set(B)] :
      ( bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),D4))
     => ( bij_betw(A,B,F3,C4,D4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B4),D4) = bot_bot(set(B)) )
           => bij_betw(A,B,F3,A5,B4) ) ) ) ) ).

% bij_betw_partition
tff(fact_7309_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B4: set(B),C4: set(A),D4: set(B)] :
      ( bij_betw(A,B,F3,A5,B4)
     => ( bij_betw(A,B,F3,C4,D4)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B4),D4) = bot_bot(set(B)) )
         => bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),D4)) ) ) ) ).

% bij_betw_combine
tff(fact_7310_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F12: fun(A,B),A15: set(A),A17: set(B),F3: fun(C,A),A5: set(C)] :
      ( bij_betw(A,B,F12,A15,A17)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F3),A5)),A15))
       => ( bij_betw(C,A,F3,A5,A15)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F12),F3),A5,A17) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_7311_disjnt__subset1,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),Z4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z4),X6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Z4),Y6)) ) ) ).

% disjnt_subset1
tff(fact_7312_disjnt__subset2,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),Z4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z4),Y6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Z4)) ) ) ).

% disjnt_subset2
tff(fact_7313_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),A15: set(B),B4: set(A),B14: set(B)] :
      ( bij_betw(A,B,F3,A5,A15)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => ( ( aa(set(A),set(B),image2(A,B,F3),B4) = B14 )
         => bij_betw(A,B,F3,B4,B14) ) ) ) ).

% bij_betw_subset
tff(fact_7314_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F12: fun(B,A),F3: fun(A,B),A15: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => ( aa(B,A,F12,aa(A,B,F3,X3)) = X3 ) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A15))
           => ( aa(A,B,F3,aa(B,A,F12,X3)) = X3 ) )
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),A15))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F12),A15)),A5))
           => bij_betw(A,B,F3,A5,A15) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_7315_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B4: set(B),G3: fun(B,A)] :
      ( inj_on(A,B,F3,A5)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B4))
       => ( inj_on(B,A,G3,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G3),B4)),A5))
           => ? [H4: fun(A,B)] : bij_betw(A,B,H4,A5,B4) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_7316_disjoint__UN__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: fun(B,set(A)),I5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),I5))))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),aa(B,set(A),B4,X4))) ) ) ).

% disjoint_UN_iff
tff(fact_7317_disjnt__insert,axiom,
    ! [A: $tType,X: A,N7: set(A),M6: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),N7))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),M6),N7))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),M6)),N7)) ) ) ).

% disjnt_insert
tff(fact_7318_bij__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N2: nat] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_7319_disjnt__sym,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),B4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B4),A5)) ) ).

% disjnt_sym
tff(fact_7320_disjnt__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),B4))
    <=> ! [X4: A] :
          ~ ( pp(aa(set(A),bool,member(A,X4),A5))
            & pp(aa(set(A),bool,member(A,X4),B4)) ) ) ).

% disjnt_iff
tff(fact_7321_bij__betwI_H,axiom,
    ! [A: $tType,B: $tType,X6: set(A),F3: fun(A,B),Y6: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),X6))
         => ! [Y3: A] :
              ( pp(aa(set(A),bool,member(A,Y3),X6))
             => ( ( aa(A,B,F3,X3) = aa(A,B,F3,Y3) )
              <=> ( X3 = Y3 ) ) ) )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),X6))
           => pp(aa(set(B),bool,member(B,aa(A,B,F3,X3)),Y6)) )
       => ( ! [Y3: B] :
              ( pp(aa(set(B),bool,member(B,Y3),Y6))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),X6))
                  & ( Y3 = aa(A,B,F3,X5) ) ) )
         => bij_betw(A,B,F3,X6,Y6) ) ) ) ).

% bij_betwI'
tff(fact_7322_bij__betw__funpow,axiom,
    ! [A: $tType,F3: fun(A,A),S: set(A),N2: nat] :
      ( bij_betw(A,A,F3,S,S)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),F3),S,S) ) ).

% bij_betw_funpow
tff(fact_7323_prod_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [H: fun(B,C),S: set(B),T2: set(C),G3: fun(C,A)] :
          ( bij_betw(B,C,H,S,T2)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_ano(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3)),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G3),T2) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_7324_sum_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [H: fun(B,C),S: set(B),T2: set(C),G3: fun(C,A)] :
          ( bij_betw(B,C,H,S,T2)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_anp(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3)),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G3),T2) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_7325_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),C4: set(B),G3: fun(A,B),B4: set(A),D4: set(B)] :
      ( bij_betw(A,B,F3,A5,C4)
     => ( bij_betw(A,B,G3,B4,D4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C4),D4) = bot_bot(set(B)) )
           => bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_aex(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F3),A5),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C4),D4)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7326_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),F3: fun(B,C),A15: fun(A,set(C))] :
      ( ! [I3: A,J2: A] :
          ( pp(aa(set(A),bool,member(A,I3),I5))
         => ( pp(aa(set(A),bool,member(A,J2),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A5,I3)),aa(A,set(B),A5,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A5,J2)),aa(A,set(B),A5,I3))) ) ) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => bij_betw(B,C,F3,aa(A,set(B),A5,I3),aa(A,set(C),A15,I3)) )
       => bij_betw(B,C,F3,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A15),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_7327_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A5: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_7328_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A5: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_7329_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y6: set(A),X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),Y6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),Y6)),X3)) )
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6)) ) ) ) ).

% disjnt_ge_max
tff(fact_7330_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B4: set(B),A5: set(A)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B4)),A5))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ).

% vimage_subset_eq
tff(fact_7331_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T8: set(C),H: fun(B,C),S: set(B),T2: set(C),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( pp(aa(set(C),bool,finite_finite2(C),T8))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T2),T8))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,member(B,A6),S3))
                   => ( aa(C,A,G3,aa(B,C,H,A6)) = zero_zero(A) ) )
               => ( ! [B5: C] :
                      ( pp(aa(set(C),bool,member(C,B5),T8))
                     => ( aa(C,A,G3,B5) = zero_zero(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_anp(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3)),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G3),T2) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_7332_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),T8: set(C),H: fun(B,C),S: set(B),T2: set(C),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( pp(aa(set(C),bool,finite_finite2(C),T8))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T2),T8))
             => ( ! [A6: B] :
                    ( pp(aa(set(B),bool,member(B,A6),S3))
                   => ( aa(C,A,G3,aa(B,C,H,A6)) = one_one(A) ) )
               => ( ! [B5: C] :
                      ( pp(aa(set(C),bool,member(C,B5),T8))
                     => ( aa(C,A,G3,B5) = one_one(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_ano(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3)),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G3),T2) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_7333_bij__image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),B4: fun(C,set(A)),A5: set(C)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B4),A5))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_aey(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F3),B4)),A5)) ) ) ).

% bij_image_INT
tff(fact_7334_card__Un__disjnt,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),B4))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_7335_sum__card__image,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pairwise(A,aTP_Lamp_anq(fun(A,set(B)),fun(A,fun(A,bool)),F3),A5)
       => ( aa(set(set(B)),nat,aa(fun(set(B),nat),fun(set(set(B)),nat),groups7311177749621191930dd_sum(set(B),nat),finite_card(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),A5)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_me(fun(A,set(B)),fun(A,nat),F3)),A5) ) ) ) ).

% sum_card_image
tff(fact_7336_init__seg__of__def,axiom,
    ! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),bool),aTP_Lamp_anr(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).

% init_seg_of_def
tff(fact_7337_bij__betw__Suc,axiom,
    ! [M6: set(nat),N7: set(nat)] :
      ( bij_betw(nat,nat,suc,M6,N7)
    <=> ( aa(set(nat),set(nat),image2(nat,nat,suc),M6) = N7 ) ) ).

% bij_betw_Suc
tff(fact_7338_bij__swap,axiom,
    ! [A: $tType,B: $tType] : bij_betw(product_prod(A,B),product_prod(B,A),product_swap(A,B),top_top(set(product_prod(A,B))),top_top(set(product_prod(B,A)))) ).

% bij_swap
tff(fact_7339_refl__on__init__seg__of,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R3)),init_seg_of(A))) ).

% refl_on_init_seg_of
tff(fact_7340_antisym__init__seg__of,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),S2)),init_seg_of(A)))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S2),R3)),init_seg_of(A)))
       => ( R3 = S2 ) ) ) ).

% antisym_init_seg_of
tff(fact_7341_trans__init__seg__of,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A)),T6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),S2)),init_seg_of(A)))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S2),T6)),init_seg_of(A)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),T6)),init_seg_of(A))) ) ) ).

% trans_init_seg_of
tff(fact_7342_pairwise__trivial,axiom,
    ! [A: $tType,I5: set(A)] : pairwise(A,aTP_Lamp_sp(A,fun(A,bool)),I5) ).

% pairwise_trivial
tff(fact_7343_pairwiseD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),S: set(A),X: A,Y: A] :
      ( pairwise(A,R2,S)
     => ( pp(aa(set(A),bool,member(A,X),S))
       => ( pp(aa(set(A),bool,member(A,Y),S))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_7344_pairwiseI,axiom,
    ! [A: $tType,S: set(A),R2: fun(A,fun(A,bool))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(A),bool,member(A,X3),S))
         => ( pp(aa(set(A),bool,member(A,Y3),S))
           => ( ( X3 != Y3 )
             => pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y3)) ) ) )
     => pairwise(A,R2,S) ) ).

% pairwiseI
tff(fact_7345_pairwise__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),S: set(A)] :
      ( pairwise(A,R2,S)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),S))
             => ( ( X4 != Xa )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),Xa)) ) ) ) ) ).

% pairwise_def
tff(fact_7346_pairwise__insert,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),X: A,S2: set(A)] :
      ( pairwise(A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S2))
    <=> ( ! [Y5: A] :
            ( ( pp(aa(set(A),bool,member(A,Y5),S2))
              & ( Y5 != X ) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R3,X),Y5))
              & pp(aa(A,bool,aa(A,fun(A,bool),R3,Y5),X)) ) )
        & pairwise(A,R3,S2) ) ) ).

% pairwise_insert
tff(fact_7347_pairwise__imageI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B),P: fun(B,fun(B,bool))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => ( pp(aa(set(A),bool,member(A,Y3),A5))
           => ( ( X3 != Y3 )
             => ( ( aa(A,B,F3,X3) != aa(A,B,F3,Y3) )
               => pp(aa(B,bool,aa(B,fun(B,bool),P,aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) ) )
     => pairwise(B,P,aa(set(A),set(B),image2(A,B,F3),A5)) ) ).

% pairwise_imageI
tff(fact_7348_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(A,bool)),F3: fun(B,A),S2: set(B)] :
      ( pairwise(A,R3,aa(set(B),set(A),image2(B,A,F3),S2))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_ans(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R3),F3),S2) ) ).

% pairwise_image
tff(fact_7349_pairwise__mono,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),A5: set(A),Q: fun(A,fun(A,bool)),B4: set(A)] :
      ( pairwise(A,P,A5)
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y3))
           => pp(aa(A,bool,aa(A,fun(A,bool),Q,X3),Y3)) )
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
         => pairwise(A,Q,B4) ) ) ) ).

% pairwise_mono
tff(fact_7350_pairwise__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),S: set(A),T2: set(A)] :
      ( pairwise(A,P,S)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
       => pairwise(A,P,T2) ) ) ).

% pairwise_subset
tff(fact_7351_pairwise__singleton,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),A5: A] : pairwise(A,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),bot_bot(set(A)))) ).

% pairwise_singleton
tff(fact_7352_pairwise__empty,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : pairwise(A,P,bot_bot(set(A))) ).

% pairwise_empty
tff(fact_7353_initial__segment__of__Diff,axiom,
    ! [A: $tType,P3: set(product_prod(A,A)),Q3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P3),Q3)),init_seg_of(A)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),P3),S2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Q3),S2))),init_seg_of(A))) ) ).

% initial_segment_of_Diff
tff(fact_7354_pairwise__alt,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),S: set(A)] :
      ( pairwise(A,R2,S)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),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)),insert(A),X4),bot_bot(set(A))))))
             => pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),Xa)) ) ) ) ).

% pairwise_alt
tff(fact_7355_disjoint__image__subset,axiom,
    ! [A: $tType,A18: set(set(A)),F3: fun(set(A),set(A))] :
      ( pairwise(set(A),disjnt(A),A18)
     => ( ! [X9: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X9),A18))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),F3,X9)),X9)) )
       => pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),image2(set(A),set(A),F3),A18)) ) ) ).

% disjoint_image_subset
tff(fact_7356_Chains__init__seg__of__Union,axiom,
    ! [A: $tType,R2: set(set(product_prod(A,A))),R3: set(product_prod(A,A))] :
      ( pp(aa(set(set(set(product_prod(A,A)))),bool,member(set(set(product_prod(A,A))),R2),chains(set(product_prod(A,A)),init_seg_of(A))))
     => ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),R3),R2))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R2))),init_seg_of(A))) ) ) ).

% Chains_init_seg_of_Union
tff(fact_7357_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(A,B),P: fun(A,bool),A3: A] :
          ( inj_on(A,B,F3,aa(fun(A,bool),set(A),collect(A),P))
         => ( pp(aa(A,bool,P,A3))
           => ( ! [Y3: A] :
                  ( pp(aa(A,bool,P,Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),aa(A,B,F3,Y3))) )
             => ( lattices_ord_arg_min(A,B,F3,P) = A3 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_7358_arg__min__nat__le,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,X))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,X))) ) ).

% arg_min_nat_le
tff(fact_7359_arg__min__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ( pp(aa(A,bool,P,lattices_ord_arg_min(A,nat,M,P)))
        & ! [Y4: A] :
            ( pp(aa(A,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,Y4))) ) ) ) ).

% arg_min_nat_lemma
tff(fact_7360_arg__min__equality,axiom,
    ! [A: $tType,C: $tType] :
      ( order(A)
     => ! [P: fun(C,bool),K: C,F3: fun(C,A)] :
          ( pp(aa(C,bool,P,K))
         => ( ! [X3: C] :
                ( pp(aa(C,bool,P,X3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,K)),aa(C,A,F3,X3))) )
           => ( aa(C,A,F3,lattices_ord_arg_min(C,A,F3,P)) = aa(C,A,F3,K) ) ) ) ) ).

% arg_min_equality
tff(fact_7361_arg__minI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P: fun(A,bool),X: A,F3: fun(A,B),Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y3)),aa(A,B,F3,X))) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y4: A] :
                        ( pp(aa(A,bool,P,Y4))
                       => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X3))) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,lattices_ord_arg_min(A,B,F3,P))) ) ) ) ) ).

% arg_minI
tff(fact_7362_mono__Chains,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R3)),chains(A,S2))) ) ).

% mono_Chains
tff(fact_7363_Chains__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : chains(A,R3) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ant(set(product_prod(A,A)),fun(set(A),bool),R3)) ).

% Chains_def
tff(fact_7364_Chains__relation__of,axiom,
    ! [A: $tType,C4: set(A),P: fun(A,fun(A,bool)),A5: set(A)] :
      ( pp(aa(set(set(A)),bool,member(set(A),C4),chains(A,order_relation_of(A,P,A5))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A5)) ) ).

% Chains_relation_of
tff(fact_7365_Chains__inits__DiffI,axiom,
    ! [A: $tType,R2: set(set(product_prod(A,A))),S2: set(product_prod(A,A))] :
      ( pp(aa(set(set(set(product_prod(A,A)))),bool,member(set(set(product_prod(A,A))),R2),chains(set(product_prod(A,A)),init_seg_of(A))))
     => pp(aa(set(set(set(product_prod(A,A)))),bool,member(set(set(product_prod(A,A))),aa(fun(set(product_prod(A,A)),bool),set(set(product_prod(A,A))),collect(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_anu(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),R2),S2))),chains(set(product_prod(A,A)),init_seg_of(A)))) ) ).

% Chains_inits_DiffI
tff(fact_7366_arg__min__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: fun(B,A),S: set(B)] : lattic7623131987881927897min_on(B,A,F3,S) = lattices_ord_arg_min(B,A,F3,aTP_Lamp_anv(set(B),fun(B,bool),S)) ) ).

% arg_min_on_def
tff(fact_7367_Chains__subset_H,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R3)
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)))),chains(A,R3))) ) ).

% Chains_subset'
tff(fact_7368_Chains__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R3)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))))) ).

% Chains_subset
tff(fact_7369_arg__min__natI,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => pp(aa(A,bool,P,lattices_ord_arg_min(A,nat,M,P))) ) ).

% arg_min_natI
tff(fact_7370_chains__alt__def,axiom,
    ! [A: $tType,A5: set(set(A))] : chains2(A,A5) = aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),A5,ord_less(set(A)))) ).

% chains_alt_def
tff(fact_7371_pred__on_Ochain__total,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C4: set(A),X: A,Y: A] :
      ( pp(aa(set(A),bool,pred_chain(A,A5,P),C4))
     => ( pp(aa(set(A),bool,member(A,X),C4))
       => ( pp(aa(set(A),bool,member(A,Y),C4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X),Y))
            | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),Y),X)) ) ) ) ) ).

% pred_on.chain_total
tff(fact_7372_subset_Ochain__total,axiom,
    ! [A: $tType,A5: set(set(A)),C4: set(set(A)),X: set(A),Y: set(A)] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C4))
     => ( pp(aa(set(set(A)),bool,member(set(A),X),C4))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y),C4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X),Y))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y),X)) ) ) ) ) ).

% subset.chain_total
tff(fact_7373_subset_OchainI,axiom,
    ! [A: $tType,C4: set(set(A)),A5: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C4),A5))
     => ( ! [X3: set(A),Y3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),C4))
           => ( pp(aa(set(set(A)),bool,member(set(A),Y3),C4))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X3),Y3))
                | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y3),X3)) ) ) )
       => pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C4)) ) ) ).

% subset.chainI
tff(fact_7374_subset_Ochain__def,axiom,
    ! [A: $tType,A5: set(set(A)),C4: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C4))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C4),A5))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),C4))
           => ! [Xa: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),Xa),C4))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X4),Xa))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Xa),X4)) ) ) ) ) ) ).

% subset.chain_def
tff(fact_7375_pred__on_Ochain__def,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C4: set(A)] :
      ( pp(aa(set(A),bool,pred_chain(A,A5,P),C4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A5))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),C4))
           => ! [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),C4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X4),Xa))
                  | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),Xa),X4)) ) ) ) ) ) ).

% pred_on.chain_def
tff(fact_7376_pred__on_OchainI,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),P: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A5))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(A),bool,member(A,X3),C4))
           => ( pp(aa(set(A),bool,member(A,Y3),C4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X3),Y3))
                | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),Y3),X3)) ) ) )
       => pp(aa(set(A),bool,pred_chain(A,A5,P),C4)) ) ) ).

% pred_on.chainI
tff(fact_7377_subset__Zorn,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ! [C6: set(set(A))] :
          ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
         => ? [X5: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X5),A5))
              & ! [Xa4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),Xa4),C6))
                 => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa4),X5)) ) ) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A5))
          & ! [Xa3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa3),A5))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa3))
               => ( Xa3 = X3 ) ) ) ) ) ).

% subset_Zorn
tff(fact_7378_subset_Ochain__empty,axiom,
    ! [A: $tType,A5: set(set(A))] : pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),bot_bot(set(set(A))))) ).

% subset.chain_empty
tff(fact_7379_pred__on_Ochain__empty,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,fun(A,bool))] : pp(aa(set(A),bool,pred_chain(A,A5,P),bot_bot(set(A)))) ).

% pred_on.chain_empty
tff(fact_7380_subset__Zorn_H,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ! [C6: set(set(A))] :
          ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)),A5)) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A5))
          & ! [Xa3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa3),A5))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa3))
               => ( Xa3 = X3 ) ) ) ) ) ).

% subset_Zorn'
tff(fact_7381_subset__chain__def,axiom,
    ! [A: $tType,A18: set(set(A)),C10: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),C10))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C10),A18))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),C10))
           => ! [Xa: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),Xa),C10))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa),X4)) ) ) ) ) ) ).

% subset_chain_def
tff(fact_7382_subset__chain__insert,axiom,
    ! [A: $tType,A18: set(set(A)),B4: set(A),B12: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),B4),B12)))
    <=> ( pp(aa(set(set(A)),bool,member(set(A),B4),A18))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),B12))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),B4))
              | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),X4)) ) )
        & pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12)) ) ) ).

% subset_chain_insert
tff(fact_7383_chain__subset__alt__def,axiom,
    ! [A: $tType,C4: set(set(A))] :
      ( chain_subset(A,C4)
    <=> pp(aa(set(set(A)),bool,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C4)) ) ).

% chain_subset_alt_def
tff(fact_7384_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A18: set(set(A))] :
      ( ( A18 != bot_bot(set(set(A))) )
     => ( ! [C11: set(set(A))] :
            ( ( C11 != bot_bot(set(set(A))) )
           => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),C11))
             => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C11)),A18)) ) )
       => ? [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),A18))
            & ! [Xa3: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),Xa3),A18))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa3))
                 => ( Xa3 = X3 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_7385_Union__in__chain,axiom,
    ! [A: $tType,B12: set(set(A)),A18: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B12))
     => ( ( B12 != bot_bot(set(set(A))) )
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)),B12)) ) ) ) ).

% Union_in_chain
tff(fact_7386_Inter__in__chain,axiom,
    ! [A: $tType,B12: set(set(A)),A18: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B12))
     => ( ( B12 != bot_bot(set(set(A))) )
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)),B12)) ) ) ) ).

% Inter_in_chain
tff(fact_7387_subset_Ochain__extend,axiom,
    ! [A: $tType,A5: set(set(A)),C4: set(set(A)),Z2: set(A)] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C4))
     => ( pp(aa(set(set(A)),bool,member(set(A),Z2),A5))
       => ( ! [X3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X3),C4))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X3),Z2)) )
         => pp(aa(set(set(A)),bool,pred_chain(set(A),A5,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))),insert(set(A)),Z2),bot_bot(set(set(A))))),C4))) ) ) ) ).

% subset.chain_extend
tff(fact_7388_pred__on_Ochain__extend,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C4: set(A),Z2: A] :
      ( pp(aa(set(A),bool,pred_chain(A,A5,P),C4))
     => ( pp(aa(set(A),bool,member(A,Z2),A5))
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),C4))
             => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X3),Z2)) )
         => pp(aa(set(A),bool,pred_chain(A,A5,P),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z2),bot_bot(set(A)))),C4))) ) ) ) ).

% pred_on.chain_extend
tff(fact_7389_finite__subset__Union__chain,axiom,
    ! [A: $tType,A5: set(A),B12: set(set(A)),A18: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)))
       => ( ( B12 != bot_bot(set(set(A))) )
         => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12))
           => ~ ! [B10: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),B10),B12))
                 => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B10)) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_7390_Chains__alt__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R3)
     => ( chains(A,R3) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))) ) ) ).

% Chains_alt_def
tff(fact_7391_multiset_Odomain,axiom,
    ! [C: $tType,B: $tType,T2: fun(C,fun(B,bool)),X5: fun(C,nat)] :
      ( pp(aa(fun(C,nat),bool,aa(fun(fun(C,nat),fun(multiset(B),bool)),fun(fun(C,nat),bool),domainp(fun(C,nat),multiset(B)),pcr_multiset(C,B,T2)),X5))
    <=> ? [Y5: fun(B,nat)] :
          ( pp(aa(fun(B,nat),bool,aa(fun(C,nat),fun(fun(B,nat),bool),bNF_rel_fun(C,B,nat,nat,T2,fequal(nat)),X5),Y5))
          & pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_anw(fun(B,nat),fun(B,bool),Y5)))) ) ) ).

% multiset.domain
tff(fact_7392_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_7393_DomainpE,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R3),A3))
     => ~ ! [B5: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B5)) ) ).

% DomainpE
tff(fact_7394_DomainPI,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B2))
     => pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R3),A3)) ) ).

% DomainPI
tff(fact_7395_Domainp_Osimps,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R3),A3))
    <=> ? [A8: A,B13: B] :
          ( ( A3 = A8 )
          & pp(aa(B,bool,aa(A,fun(B,bool),R3,A8),B13)) ) ) ).

% Domainp.simps
tff(fact_7396_Domainp_Ocases,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R3),A3))
     => ~ ! [B5: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B5)) ) ).

% Domainp.cases
tff(fact_7397_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),X)),aa(A,A,aa(A,fun(A,A),Disj,Z2),X)) ) ) ).

% abstract_boolean_algebra.disj_conj_distrib2
tff(fact_7398_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).

% abstract_boolean_algebra.conj_disj_distrib2
tff(fact_7399_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,Compl,X) = aa(A,A,Compl,Y) )
      <=> ( X = Y ) ) ) ).

% abstract_boolean_algebra.compl_eq_compl_iff
tff(fact_7400_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,X),Z2)) ) ) ).

% abstract_boolean_algebra.disj_conj_distrib
tff(fact_7401_abstract__boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,Compl,X)) = One ) ) ).

% abstract_boolean_algebra.disj_cancel_right
tff(fact_7402_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).

% abstract_boolean_algebra.conj_disj_distrib
tff(fact_7403_abstract__boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,X)) = Zero ) ) ).

% abstract_boolean_algebra.conj_cancel_right
tff(fact_7404_abstract__boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,A3: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),X) = Zero )
       => ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),X) = One )
         => ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),Y) = Zero )
           => ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),Y) = One )
             => ( X = Y ) ) ) ) ) ) ).

% abstract_boolean_algebra.complement_unique
tff(fact_7405_abstract__boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),X) = One ) ) ).

% abstract_boolean_algebra.disj_cancel_left
tff(fact_7406_abstract__boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),X) = Zero ) ) ).

% abstract_boolean_algebra.conj_cancel_left
tff(fact_7407_abstract__boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),Zero) = X ) ) ).

% abstract_boolean_algebra.disj_zero_right
tff(fact_7408_abstract__boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),Zero) = Zero ) ) ).

% abstract_boolean_algebra.conj_zero_right
tff(fact_7409_abstract__boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,X),One) = One ) ) ).

% abstract_boolean_algebra.disj_one_right
tff(fact_7410_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Disj,X),Y)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).

% abstract_boolean_algebra.de_Morgan_disj
tff(fact_7411_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Conj,X),Y)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).

% abstract_boolean_algebra.de_Morgan_conj
tff(fact_7412_abstract__boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Zero),X) = Zero ) ) ).

% abstract_boolean_algebra.conj_zero_left
tff(fact_7413_abstract__boolean__algebra_Oconj__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),One) = X ) ) ).

% abstract_boolean_algebra.conj_one_right
tff(fact_7414_abstract__boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,One),X) = One ) ) ).

% abstract_boolean_algebra.disj_one_left
tff(fact_7415_abstract__boolean__algebra_Odouble__compl,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,Compl,X)) = X ) ) ).

% abstract_boolean_algebra.double_compl
tff(fact_7416_abstract__boolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,aa(A,fun(A,A),Conj,X),Y) = Zero )
       => ( ( aa(A,A,aa(A,fun(A,A),Disj,X),Y) = One )
         => ( aa(A,A,Compl,X) = Y ) ) ) ) ).

% abstract_boolean_algebra.compl_unique
tff(fact_7417_abstract__boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,Zero) = One ) ) ).

% abstract_boolean_algebra.compl_zero
tff(fact_7418_abstract__boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,One) = Zero ) ) ).

% abstract_boolean_algebra.compl_one
tff(fact_7419_multiset_Odomain__eq,axiom,
    ! [A: $tType,X5: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(fun(A,nat),fun(multiset(A),bool)),fun(fun(A,nat),bool),domainp(fun(A,nat),multiset(A)),pcr_multiset(A,A,fequal(A))),X5))
    <=> pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_gi(fun(A,nat),fun(A,bool),X5)))) ) ).

% multiset.domain_eq
tff(fact_7420_multiset_Odomain__par__left__total,axiom,
    ! [B: $tType,C: $tType,T2: fun(C,fun(B,bool)),P6: fun(fun(C,nat),bool)] :
      ( left_total(fun(C,nat),fun(B,nat),bNF_rel_fun(C,B,nat,nat,T2,fequal(nat)))
     => ( pp(aa(fun(fun(B,nat),bool),bool,aa(fun(fun(C,nat),bool),fun(fun(fun(B,nat),bool),bool),bNF_rel_fun(fun(C,nat),fun(B,nat),bool,bool,bNF_rel_fun(C,B,nat,nat,T2,fequal(nat)),fequal(bool)),P6),aTP_Lamp_anx(fun(B,nat),bool)))
       => ( aa(fun(fun(C,nat),fun(multiset(B),bool)),fun(fun(C,nat),bool),domainp(fun(C,nat),multiset(B)),pcr_multiset(C,B,T2)) = P6 ) ) ) ).

% multiset.domain_par_left_total
tff(fact_7421_Rep__unit,axiom,
    ! [X: product_unit] : pp(aa(set(bool),bool,member(bool,aa(product_unit,bool,product_Rep_unit,X)),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool))))) ).

% Rep_unit
tff(fact_7422_Rep__unit__inject,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( pp(aa(product_unit,bool,product_Rep_unit,X))
      <=> pp(aa(product_unit,bool,product_Rep_unit,Y)) )
    <=> ( X = Y ) ) ).

% Rep_unit_inject
tff(fact_7423_Rep__unit__induct,axiom,
    ! [Y: bool,P: fun(bool,bool)] :
      ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ( ! [X3: product_unit] : pp(aa(bool,bool,P,aa(product_unit,bool,product_Rep_unit,X3)))
       => pp(aa(bool,bool,P,Y)) ) ) ).

% Rep_unit_induct
tff(fact_7424_Rep__unit__cases,axiom,
    ! [Y: bool] :
      ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ~ ! [X3: product_unit] :
            ( pp(Y)
          <=> ~ pp(aa(product_unit,bool,product_Rep_unit,X3)) ) ) ).

% Rep_unit_cases
tff(fact_7425_type__definition__unit,axiom,
    type_definition(product_unit,bool,product_Rep_unit,product_Abs_unit,aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))) ).

% type_definition_unit
tff(fact_7426_Abs__unit__inverse,axiom,
    ! [Y: bool] :
      ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ( pp(aa(product_unit,bool,product_Rep_unit,aa(bool,product_unit,product_Abs_unit,Y)))
      <=> pp(Y) ) ) ).

% Abs_unit_inverse
tff(fact_7427_Rep__unit__inverse,axiom,
    ! [X: product_unit] : aa(bool,product_unit,product_Abs_unit,aa(product_unit,bool,product_Rep_unit,X)) = X ).

% Rep_unit_inverse
tff(fact_7428_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: bool] :
          ( ( X = aa(bool,product_unit,product_Abs_unit,Y3) )
         => ~ pp(aa(set(bool),bool,member(bool,Y3),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool))))) ) ).

% Abs_unit_cases
tff(fact_7429_Abs__unit__induct,axiom,
    ! [P: fun(product_unit,bool),X: product_unit] :
      ( ! [Y3: bool] :
          ( pp(aa(set(bool),bool,member(bool,Y3),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
         => pp(aa(product_unit,bool,P,aa(bool,product_unit,product_Abs_unit,Y3))) )
     => pp(aa(product_unit,bool,P,X)) ) ).

% Abs_unit_induct
tff(fact_7430_Abs__unit__inject,axiom,
    ! [X: bool,Y: bool] :
      ( pp(aa(set(bool),bool,member(bool,X),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
       => ( ( aa(bool,product_unit,product_Abs_unit,X) = aa(bool,product_unit,product_Abs_unit,Y) )
        <=> ( pp(X)
          <=> pp(Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_7431_multiset_Odomain__par,axiom,
    ! [B: $tType,C: $tType,T2: fun(C,fun(B,bool)),DT: fun(C,bool),DS: fun(nat,bool),P23: fun(fun(C,nat),bool)] :
      ( ( aa(fun(C,fun(B,bool)),fun(C,bool),domainp(C,B),T2) = DT )
     => ( ( aa(fun(nat,fun(nat,bool)),fun(nat,bool),domainp(nat,nat),fequal(nat)) = DS )
       => ( left_unique(C,B,T2)
         => ( pp(aa(fun(fun(B,nat),bool),bool,aa(fun(fun(C,nat),bool),fun(fun(fun(B,nat),bool),bool),bNF_rel_fun(fun(C,nat),fun(B,nat),bool,bool,bNF_rel_fun(C,B,nat,nat,T2,fequal(nat)),fequal(bool)),P23),aTP_Lamp_anx(fun(B,nat),bool)))
           => ( aa(fun(fun(C,nat),fun(multiset(B),bool)),fun(fun(C,nat),bool),domainp(fun(C,nat),multiset(B)),pcr_multiset(C,B,T2)) = aa(fun(fun(C,nat),bool),fun(fun(C,nat),bool),aa(fun(fun(C,nat),bool),fun(fun(fun(C,nat),bool),fun(fun(C,nat),bool)),inf_inf(fun(fun(C,nat),bool)),aa(fun(nat,bool),fun(fun(C,nat),bool),basic_pred_fun(C,nat,DT),DS)),P23) ) ) ) ) ) ).

% multiset.domain_par
tff(fact_7432_mono__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType] :
      ( ( order(B)
        & order(D)
        & order(C)
        & order(A) )
     => ! [A5: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool))] :
          ( bi_total(A,B,A5)
         => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),bool),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A5,bNF_rel_fun(A,B,bool,bool,A5,fequal(bool))),ord_less_eq(A)),ord_less_eq(B)))
           => ( pp(aa(fun(D,fun(D,bool)),bool,aa(fun(C,fun(C,bool)),fun(fun(D,fun(D,bool)),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B4,bNF_rel_fun(C,D,bool,bool,B4,fequal(bool))),ord_less_eq(C)),ord_less_eq(D)))
             => pp(aa(fun(fun(B,D),bool),bool,aa(fun(fun(A,C),bool),fun(fun(fun(B,D),bool),bool),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A5,B4),fequal(bool)),order_mono(A,C)),order_mono(B,D))) ) ) ) ) ).

% mono_transfer
tff(fact_7433_typedef__left__unique,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T2: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T2,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => left_unique(A,B,T2) ) ) ).

% typedef_left_unique
tff(fact_7434_pred__fun__def,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,bool),B4: fun(B,bool),X5: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(fun(B,bool),fun(fun(A,B),bool),basic_pred_fun(A,B,A5),B4),X5))
    <=> ! [Xa: A] :
          ( pp(aa(A,bool,A5,Xa))
         => pp(aa(B,bool,B4,aa(A,B,X5,Xa))) ) ) ).

% pred_fun_def
tff(fact_7435_fun_Opred__True,axiom,
    ! [A: $tType,D: $tType,X5: fun(D,A)] : pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),aTP_Lamp_bm(A,bool)),X5)) ).

% fun.pred_True
tff(fact_7436_fun_Opred__mono,axiom,
    ! [D: $tType,A: $tType,P: fun(A,bool),Pa: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Pa))
     => pp(aa(fun(fun(D,A),bool),bool,aa(fun(fun(D,A),bool),fun(fun(fun(D,A),bool),bool),ord_less_eq(fun(fun(D,A),bool)),aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),P)),aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),Pa))) ) ).

% fun.pred_mono
tff(fact_7437_fun_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,D: $tType,X: fun(D,A),Ya: fun(D,A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),aa(fun(A,B),fun(A,bool),aTP_Lamp_anz(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),G3)),Ya))
       => ( aa(fun(D,A),fun(D,B),comp(A,B,D,F3),X) = aa(fun(D,A),fun(D,B),comp(A,B,D,G3),Ya) ) ) ) ).

% fun.map_cong_pred
tff(fact_7438_pred__fun__True__id,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: fun(B,bool),F3: fun(C,B)] :
      ( nO_MATCH(fun(A,A),fun(B,bool),id(A),P3)
     => ( pp(aa(fun(C,B),bool,aa(fun(B,bool),fun(fun(C,B),bool),basic_pred_fun(C,B,aTP_Lamp_aoa(C,bool)),P3),F3))
      <=> pp(aa(fun(C,bool),bool,aa(fun(bool,bool),fun(fun(C,bool),bool),basic_pred_fun(C,bool,aTP_Lamp_aoa(C,bool)),id(bool)),aa(fun(C,B),fun(C,bool),comp(B,bool,C,P3),F3))) ) ) ).

% pred_fun_True_id
tff(fact_7439_fun_Opred__mono__strong,axiom,
    ! [A: $tType,D: $tType,P: fun(A,bool),X: fun(D,A),Pa: fun(A,bool)] :
      ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),P),X))
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),image2(D,A,X),top_top(set(D)))))
           => ( pp(aa(A,bool,P,Z3))
             => pp(aa(A,bool,Pa,Z3)) ) )
       => pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),Pa),X)) ) ) ).

% fun.pred_mono_strong
tff(fact_7440_fun_Opred__cong,axiom,
    ! [A: $tType,D: $tType,X: fun(D,A),Ya: fun(D,A),P: fun(A,bool),Pa: fun(A,bool)] :
      ( ( X = Ya )
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),image2(D,A,Ya),top_top(set(D)))))
           => ( pp(aa(A,bool,P,Z3))
            <=> pp(aa(A,bool,Pa,Z3)) ) )
       => ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),P),X))
        <=> pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),Pa),Ya)) ) ) ) ).

% fun.pred_cong
tff(fact_7441_fun_Opred__rel,axiom,
    ! [A: $tType,D: $tType,P: fun(A,bool),X: fun(D,A)] :
      ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),P),X))
    <=> pp(aa(fun(D,A),bool,aa(fun(D,A),fun(fun(D,A),bool),bNF_rel_fun(D,D,A,A,fequal(D),bNF_eq_onp(A,P)),X),X)) ) ).

% fun.pred_rel
tff(fact_7442_fun_ODomainp__rel,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : aa(fun(fun(C,A),fun(fun(C,B),bool)),fun(fun(C,A),bool),domainp(fun(C,A),fun(C,B)),bNF_rel_fun(C,C,A,B,fequal(C),R2)) = aa(fun(A,bool),fun(fun(C,A),bool),basic_pred_fun(C,A,aTP_Lamp_aoa(C,bool)),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R2)) ).

% fun.Domainp_rel
tff(fact_7443_fun_Opred__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,Q: fun(B,bool),F3: fun(A,B),X: fun(D,A)] :
      ( pp(aa(fun(D,B),bool,aa(fun(B,bool),fun(fun(D,B),bool),basic_pred_fun(D,B,aTP_Lamp_any(D,bool)),Q),aa(fun(D,A),fun(D,B),comp(A,B,D,F3),X)))
    <=> pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),aa(fun(A,B),fun(A,bool),comp(B,bool,A,Q),F3)),X)) ) ).

% fun.pred_map
tff(fact_7444_fun_Opred__set,axiom,
    ! [D: $tType,A: $tType,P: fun(A,bool),X5: fun(D,A)] :
      ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),P),X5))
    <=> ! [Xa: A] :
          ( pp(aa(set(A),bool,member(A,Xa),aa(set(D),set(A),image2(D,A,X5),top_top(set(D)))))
         => pp(aa(A,bool,P,Xa)) ) ) ).

% fun.pred_set
tff(fact_7445_fun_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(fun(D,B),bool)),bool,aa(fun(fun(A,bool),fun(fun(D,A),bool)),fun(fun(fun(B,bool),fun(fun(D,B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(fun(D,A),bool),fun(fun(D,B),bool),bNF_rel_fun(A,B,bool,bool,R2,fequal(bool)),bNF_rel_fun(fun(D,A),fun(D,B),bool,bool,bNF_rel_fun(D,D,A,B,fequal(D),R2),fequal(bool))),basic_pred_fun(D,A,aTP_Lamp_any(D,bool))),basic_pred_fun(D,B,aTP_Lamp_any(D,bool)))) ).

% fun.pred_transfer
tff(fact_7446_fun_Orel__eq__onp,axiom,
    ! [D: $tType,A: $tType,P: fun(A,bool)] : bNF_rel_fun(D,D,A,A,fequal(D),bNF_eq_onp(A,P)) = bNF_eq_onp(fun(D,A),aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_any(D,bool)),P)) ).

% fun.rel_eq_onp
tff(fact_7447_Inter__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A5)
     => ( bi_total(A,B,A5)
       => pp(aa(fun(set(set(B)),set(B)),bool,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),bool),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A5)),bNF_rel_set(A,B,A5)),complete_Inf_Inf(set(A))),complete_Inf_Inf(set(B)))) ) ) ).

% Inter_transfer
tff(fact_7448_in__measures_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat),Fs: list(fun(A,nat))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
        | ( ( aa(A,nat,F3,X) = aa(A,nat,F3,Y) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs))) ) ) ) ).

% in_measures(2)
tff(fact_7449_in__measures_I1_J,axiom,
    ! [A: $tType,X: A,Y: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,nil(fun(A,nat))))) ).

% in_measures(1)
tff(fact_7450_subset__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A5)
     => pp(aa(fun(set(B),fun(set(B),bool)),bool,aa(fun(set(A),fun(set(A),bool)),fun(fun(set(B),fun(set(B),bool)),bool),bNF_rel_fun(set(A),set(B),fun(set(A),bool),fun(set(B),bool),bNF_rel_set(A,B,A5),bNF_rel_fun(set(A),set(B),bool,bool,bNF_rel_set(A,B,A5),fequal(bool))),ord_less_eq(set(A))),ord_less_eq(set(B)))) ) ).

% subset_transfer
tff(fact_7451_strict__subset__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A5)
     => pp(aa(fun(set(B),fun(set(B),bool)),bool,aa(fun(set(A),fun(set(A),bool)),fun(fun(set(B),fun(set(B),bool)),bool),bNF_rel_fun(set(A),set(B),fun(set(A),bool),fun(set(B),bool),bNF_rel_set(A,B,A5),bNF_rel_fun(set(A),set(B),bool,bool,bNF_rel_set(A,B,A5),fequal(bool))),ord_less(set(A))),ord_less(set(B)))) ) ).

% strict_subset_transfer
tff(fact_7452_typedef__bi__unique,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T2: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T2,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => bi_unique(A,B,T2) ) ) ).

% typedef_bi_unique
tff(fact_7453_measures__less,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs)))) ) ).

% measures_less
tff(fact_7454_measures__lesseq,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs)))) ) ) ).

% measures_lesseq
tff(fact_7455_measures__def,axiom,
    ! [A: $tType,Fs: list(fun(A,nat))] : measures(A,Fs) = inv_image(list(nat),A,lex(nat,less_than),aTP_Lamp_aob(list(fun(A,nat)),fun(A,list(nat)),Fs)) ).

% measures_def
tff(fact_7456_inter__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A5)
     => pp(aa(fun(set(B),fun(set(B),set(B))),bool,aa(fun(set(A),fun(set(A),set(A))),fun(fun(set(B),fun(set(B),set(B))),bool),bNF_rel_fun(set(A),set(B),fun(set(A),set(A)),fun(set(B),set(B)),bNF_rel_set(A,B,A5),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A5),bNF_rel_set(A,B,A5))),inf_inf(set(A))),inf_inf(set(B)))) ) ).

% inter_transfer
tff(fact_7457_right__total__Inter__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A5)
     => ( right_total(A,B,A5)
       => pp(aa(fun(set(set(B)),set(B)),bool,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),bool),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A5)),bNF_rel_set(A,B,A5)),aTP_Lamp_aoc(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),A5)),complete_Inf_Inf(set(B)))) ) ) ).

% right_total_Inter_transfer
tff(fact_7458_vimage__right__total__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,B4: fun(A,fun(B,bool)),A5: fun(C,fun(D,bool))] :
      ( bi_unique(A,B,B4)
     => ( right_total(C,D,A5)
       => pp(aa(fun(fun(D,B),fun(set(B),set(D))),bool,aa(fun(fun(C,A),fun(set(A),set(C))),fun(fun(fun(D,B),fun(set(B),set(D))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(set(A),set(C)),fun(set(B),set(D)),bNF_rel_fun(C,D,A,B,A5,B4),bNF_rel_fun(set(A),set(B),set(C),set(D),bNF_rel_set(A,B,B4),bNF_rel_set(C,D,A5))),aTP_Lamp_aod(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),A5)),vimage(D,B))) ) ) ).

% vimage_right_total_transfer
tff(fact_7459_typedef__right__total,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T2: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T2,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => right_total(A,B,T2) ) ) ).

% typedef_right_total
tff(fact_7460_right__total__Collect__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] :
      ( right_total(A,B,A5)
     => pp(aa(fun(fun(B,bool),set(B)),bool,aa(fun(fun(A,bool),set(A)),fun(fun(fun(B,bool),set(B)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),set(A),set(B),bNF_rel_fun(A,B,bool,bool,A5,fequal(bool)),bNF_rel_set(A,B,A5)),aTP_Lamp_aof(fun(A,fun(B,bool)),fun(fun(A,bool),set(A)),A5)),collect(B))) ) ).

% right_total_Collect_transfer
tff(fact_7461_right__total__Domainp__transfer,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,B4: fun(A,fun(B,bool)),A5: fun(C,fun(D,bool))] :
      ( right_total(A,B,B4)
     => pp(aa(fun(fun(D,fun(B,bool)),fun(D,bool)),bool,aa(fun(fun(C,fun(A,bool)),fun(C,bool)),fun(fun(fun(D,fun(B,bool)),fun(D,bool)),bool),bNF_rel_fun(fun(C,fun(A,bool)),fun(D,fun(B,bool)),fun(C,bool),fun(D,bool),bNF_rel_fun(C,D,fun(A,bool),fun(B,bool),A5,bNF_rel_fun(A,B,bool,bool,B4,fequal(bool))),bNF_rel_fun(C,D,bool,bool,A5,fequal(bool))),aTP_Lamp_aog(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(C,bool)),B4)),domainp(D,B))) ) ).

% right_total_Domainp_transfer
tff(fact_7462_right__total__Compl__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A5)
     => ( right_total(A,B,A5)
       => pp(aa(fun(set(B),set(B)),bool,aa(fun(set(A),set(A)),fun(fun(set(B),set(B)),bool),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A5),bNF_rel_set(A,B,A5)),aTP_Lamp_aoh(fun(A,fun(B,bool)),fun(set(A),set(A)),A5)),uminus_uminus(set(B)))) ) ) ).

% right_total_Compl_transfer
tff(fact_7463_right__total__fun__eq__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A5: fun(A,fun(B,bool)),B4: fun(C,fun(D,bool))] :
      ( right_total(A,B,A5)
     => ( bi_unique(C,D,B4)
       => pp(aa(fun(fun(B,D),fun(fun(B,D),bool)),bool,aa(fun(fun(A,C),fun(fun(A,C),bool)),fun(fun(fun(B,D),fun(fun(B,D),bool)),bool),bNF_rel_fun(fun(A,C),fun(B,D),fun(fun(A,C),bool),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A5,B4),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A5,B4),fequal(bool))),aTP_Lamp_aoi(fun(A,fun(B,bool)),fun(fun(A,C),fun(fun(A,C),bool)),A5)),fequal(fun(B,D)))) ) ) ).

% right_total_fun_eq_transfer
tff(fact_7464_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [N2: int,X: A] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
           => ( power_int(A,X,N2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(int,nat,nat2,N2)) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
           => ( power_int(A,X,N2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N2))) ) ) ) ) ).

% power_int_def
tff(fact_7465_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) )
          <=> ( A3 = B2 ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_7466_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W2: num,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W2),M)),power_int(A,Y,M)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_7467_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,W2: num,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W2)),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,aa(num,A,numeral_numeral(A),W2),M)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_7468_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)) = one_one(A) ) ).

% power_int_minus_one_mult_self
tff(fact_7469_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),B2)) = B2 ) ).

% power_int_minus_one_mult_self'
tff(fact_7470_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),power_int(A,X,aa(num,int,numeral_numeral(int),N2))) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2))) ) ).

% power_int_add_numeral
tff(fact_7471_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N2: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N2))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2)))),B2) ) ).

% power_int_add_numeral2
tff(fact_7472_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N2)),power_int(A,B2,N2)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_7473_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_7474_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N2: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N2)),power_int(A,A3,N7))) ) ) ) ).

% power_int_strict_increasing
tff(fact_7475_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N2: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,N2)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,N2)) ) ).

% power_int_commutes
tff(fact_7476_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,Y,M)) ) ).

% power_int_mult_distrib
tff(fact_7477_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),power_int(A,X,N2))) ) ) ).

% zero_less_power_int
tff(fact_7478_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N2: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N2)),power_int(A,A3,N7))) ) ) ) ).

% power_int_increasing
tff(fact_7479_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),power_int(A,X,N2))) ) ) ).

% zero_le_power_int
tff(fact_7480_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N2: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N7)),power_int(A,A3,N2))) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_7481_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N2)),power_int(A,Y,N2))) ) ) ) ) ).

% power_int_mono
tff(fact_7482_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N2),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,B2,N2)),power_int(A,A3,N2))) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_7483_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X,N2))) ) ) ) ).

% one_le_power_int
tff(fact_7484_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),power_int(A,A3,N2))) ) ) ) ).

% one_less_power_int
tff(fact_7485_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N2: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N2) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,X,N2)) ) ) ) ).

% power_int_add
tff(fact_7486_power__int__minus__left__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( division_ring(A)
        & one(B)
        & uminus(B) )
     => ! [X: C,A3: A,N2: int] :
          ( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N2)),power_int(A,A3,N2)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_7487_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N2),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,B2,N2)),power_int(A,A3,N2))) ) ) ) ) ).

% power_int_antimono
tff(fact_7488_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N2)),power_int(A,B2,N2))) ) ) ) ) ).

% power_int_strict_mono
tff(fact_7489_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N2)),one_one(A))) ) ) ) ) ).

% power_int_le_one
tff(fact_7490_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N2: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N2),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => ( ( ( A3 != zero_zero(A) )
                  | ( N7 != zero_zero(int) )
                  | ( N2 = zero_zero(int) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N7)),power_int(A,A3,N2))) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_7491_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M)),power_int(A,X,N2)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N2)) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_7492_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X,M)),power_int(A,X,N2)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N2)) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_7493_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N2: int] :
          ( ( ( X != zero_zero(A) )
            | ( N2 != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),N2),one_one(int)))),X) = power_int(A,X,N2) ) ) ) ).

% power_int_minus_mult
tff(fact_7494_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),X) ) ) ) ).

% power_int_add_1
tff(fact_7495_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M)) ) ) ) ).

% power_int_add_1'
tff(fact_7496_Zorns__po__lemma,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ! [C6: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),C6),chains(A,R3)))
           => ? [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,member(A,Xa4),C6))
                   => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X5)),R3)) ) ) )
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R3))
                 => ( Xa3 = X3 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_7497_Partial__order__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => order_7125193373082350890der_on(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% Partial_order_Restr
tff(fact_7498_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R3),id2(A))))
              | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R3),id2(A)))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( ( A3 = B2 )
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_7499_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B4: set(A),F4: fun(A,filter(B)),P: fun(B,bool)] :
      ( ( B4 != bot_bot(set(A)) )
     => ( ! [A6: A] :
            ( pp(aa(set(A),bool,member(A,A6),B4))
           => ! [B5: A] :
                ( pp(aa(set(A),bool,member(A,B5),B4))
               => ? [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),B4))
                    & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F4,X5)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A6)),aa(A,filter(B),F4,B5)))) ) ) )
       => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B4)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),B4))
              & eventually(B,P,aa(A,filter(B),F4,X4)) ) ) ) ) ).

% eventually_INF_base
tff(fact_7500_eventually__top,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( eventually(A,P,top_top(filter(A)))
    <=> ! [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).

% eventually_top
tff(fact_7501_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P: bool] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_oz(bool,fun(A,bool),P),F4)
      <=> pp(P) ) ) ).

% eventually_const
tff(fact_7502_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_aki(fun(nat,bool),fun(nat,bool),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_7503_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,bool),aTP_Lamp_aoj(fun(nat,bool),fun(nat,fun(nat,bool)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_7504_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A5: set(A),P: fun(set(A),bool)] :
      ( ! [X9: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X9))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X9),A5))
           => pp(aa(set(A),bool,P,X9)) ) )
     => eventually(set(A),P,finite5375528669736107172at_top(A,A5)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_7505_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F3: fun(A,B),P: fun(B,bool),G3: fun(B,A)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G3,X3))) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F3,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_7506_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),aTP_Lamp_se(A,fun(A,bool)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_7507_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N12: A] :
            ! [N: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),N12))
             => pp(aa(A,bool,P,N)) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_7508_le__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F5: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F5))
    <=> ! [P8: fun(A,bool)] :
          ( eventually(A,P8,F5)
         => eventually(A,P8,F4) ) ) ).

% le_filter_def
tff(fact_7509_filter__leI,axiom,
    ! [A: $tType,F5: filter(A),F4: filter(A)] :
      ( ! [P2: fun(A,bool)] :
          ( eventually(A,P2,F5)
         => eventually(A,P2,F4) )
     => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F5)) ) ).

% filter_leI
tff(fact_7510_filter__leD,axiom,
    ! [A: $tType,F4: filter(A),F5: filter(A),P: fun(A,bool)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F5))
     => ( eventually(A,P,F5)
       => eventually(A,P,F4) ) ) ).

% filter_leD
tff(fact_7511_le__principal,axiom,
    ! [A: $tType,F4: filter(A),A5: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),principal(A,A5)))
    <=> eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A5),F4) ) ).

% le_principal
tff(fact_7512_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_7513_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( pp(aa(filter(nat),bool,aa(filter(nat),fun(filter(nat),bool),ord_less_eq(filter(nat)),F4),at_top(nat)))
    <=> ! [N12: nat] : eventually(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N12),F4) ) ).

% le_sequentially
tff(fact_7514_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),X3))
             => pp(aa(A,bool,P,X3)) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_7515_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N12: A] :
            ! [N: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N12),N))
             => pp(aa(A,bool,P,N)) ) ) ) ).

% eventually_at_top_linorder
tff(fact_7516_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,bool)] :
      ( ! [X3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),X3))
         => pp(aa(nat,bool,P,X3)) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_7517_eventually__sequentially,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N12: nat] :
        ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N12),N))
         => pp(aa(nat,bool,P,N)) ) ) ).

% eventually_sequentially
tff(fact_7518_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(B),G4: filter(A),F5: filter(B),G8: filter(A),F12: fun(A,B)] :
      ( filterlim(A,B,F3,F4,G4)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F4),F5))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G8),G4))
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_anz(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),F12),G8)
           => filterlim(A,B,F12,F5,G8) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_7519_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P: fun(set(A),bool),A5: set(A)] :
      ( eventually(set(A),P,finite5375528669736107172at_top(A,A5))
    <=> ? [X14: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X14))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X14),A5))
          & ! [Y9: set(A)] :
              ( ( pp(aa(set(A),bool,finite_finite2(A),Y9))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X14),Y9))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y9),A5)) )
             => pp(aa(set(A),bool,P,Y9)) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_7520_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_aaf(fun(A,fun(B,bool)),fun(A,bool),P),F4)
    <=> ? [Y9: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_aok(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),P),Y9),F4) ) ).

% eventually_ex
tff(fact_7521_eventually__sup,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),F5: filter(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F5))
    <=> ( eventually(A,P,F4)
        & eventually(A,P,F5) ) ) ).

% eventually_sup
tff(fact_7522_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F4: filter(A),F3: fun(B,A),G4: filter(B)] :
      ( eventually(A,P,F4)
     => ( filterlim(B,A,F3,F4,G4)
       => eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aol(fun(A,bool),fun(fun(B,A),fun(B,bool)),P),F3),G4) ) ) ).

% eventually_compose_filterlim
tff(fact_7523_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F14: filter(A),F15: filter(A),F23: filter(B),F24: filter(B),F3: fun(B,A),G3: fun(B,A)] :
      ( ( F14 = F15 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aom(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G3),F23)
         => ( filterlim(B,A,F3,F14,F23)
          <=> filterlim(B,A,G3,F15,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_7524_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F23: filter(B),F14: filter(A)] :
      ( filterlim(A,B,F3,F23,F14)
    <=> ! [P8: fun(B,bool)] :
          ( eventually(B,P8,F23)
         => eventually(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_acp(fun(A,B),fun(fun(B,bool),fun(A,bool)),F3),P8),F14) ) ) ).

% filterlim_iff
tff(fact_7525_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),S: set(B),F4: filter(A)] :
      ( filterlim(A,B,F3,principal(B,S),F4)
    <=> eventually(A,aa(set(B),fun(A,bool),aTP_Lamp_acs(fun(A,B),fun(set(B),fun(A,bool)),F3),S),F4) ) ).

% filterlim_principal
tff(fact_7526_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N12: A] :
            ! [N: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N12),N))
             => pp(aa(A,bool,P,N)) ) ) ) ).

% eventually_at_top_dense
tff(fact_7527_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_7528_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_lm(nat,bool),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_7529_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_aon(A,fun(A,bool),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_7530_eventually__finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A5: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( eventually(set(A),P,finite5375528669736107172at_top(A,A5))
      <=> pp(aa(set(A),bool,P,A5)) ) ) ).

% eventually_finite_subsets_at_top_finite
tff(fact_7531_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
         => ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = B2 ) )
        & ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
         => ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = A3 ) ) ) ) ).

% wo_rel.max2_def
tff(fact_7532_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( ( ( Y4 != X3 )
                  & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R3)) )
               => pp(aa(A,bool,P,Y4)) )
           => pp(aa(A,bool,P,X3)) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% wo_rel.well_order_induct
tff(fact_7533_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ! [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R3))
                | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),R3)) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_7534_well__order__induct__imp,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( ( ( Y4 != X3 )
                  & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R3)) )
               => ( pp(aa(set(A),bool,member(A,Y4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
                 => pp(aa(A,bool,P,Y4)) ) )
           => ( pp(aa(set(A),bool,member(A,X3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
             => pp(aa(A,bool,P,X3)) ) )
       => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% well_order_induct_imp
tff(fact_7535_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = A3 )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3)) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_7536_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = B2 )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_7537_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3)) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_7538_eventually__all__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(B)
     => ! [P: fun(A,fun(B,bool)),Net: filter(A)] :
          ( ! [Y3: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_aoo(fun(A,fun(B,bool)),fun(B,fun(A,bool)),P),Y3),Net)
         => eventually(A,aTP_Lamp_aop(fun(A,fun(B,bool)),fun(A,bool),P),Net) ) ) ).

% eventually_all_finite
tff(fact_7539_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_aoq(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_7540_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N12: A] :
            ! [N: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),N12))
             => pp(aa(A,bool,P,N)) ) ) ) ).

% eventually_at_bot_dense
tff(fact_7541_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F4)
     => ( ~ eventually(A,Q,F4)
       => ~ eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_cg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F4) ) ) ).

% not_eventually_impI
tff(fact_7542_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ag(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F4)
    <=> ( eventually(A,P,F4)
        & eventually(A,Q,F4) ) ) ).

% eventually_conj_iff
tff(fact_7543_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F4)
     => ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_cg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_rev_mp
tff(fact_7544_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aor(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F4)
     => ( eventually(A,P,F4)
      <=> eventually(A,Q,F4) ) ) ).

% eventually_subst
tff(fact_7545_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),Q: fun(A,bool),R2: fun(A,bool)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => ( ! [I3: A] :
              ( pp(aa(A,bool,P,I3))
             => ( pp(aa(A,bool,Q,I3))
               => pp(aa(A,bool,R2,I3)) ) )
         => eventually(A,R2,F4) ) ) ) ).

% eventually_elim2
tff(fact_7546_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ag(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F4) ) ) ).

% eventually_conj
tff(fact_7547_eventually__True,axiom,
    ! [A: $tType,F4: filter(A)] : eventually(A,aTP_Lamp_bm(A,bool),F4) ).

% eventually_True
tff(fact_7548_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_cg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F4)
     => ( eventually(A,P,F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_mp
tff(fact_7549_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,bool),C4: bool,F4: filter(A)] :
      ( eventually(A,aa(bool,fun(A,bool),aTP_Lamp_aos(fun(A,bool),fun(bool,fun(A,bool)),P),C4),F4)
    <=> ( eventually(A,P,F4)
        | pp(C4) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_7550_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C4: bool,P: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aot(bool,fun(fun(A,bool),fun(A,bool)),C4),P),F4)
    <=> ( pp(C4)
        | eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_7551_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C4: bool,P: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aou(bool,fun(fun(A,bool),fun(A,bool)),C4),P),F4)
    <=> ( pp(C4)
       => eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_7552_eventuallyI,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A)] :
      ( ! [X3: A] : pp(aa(A,bool,P,X3))
     => eventually(A,P,F4) ) ).

% eventuallyI
tff(fact_7553_filter__eq__iff,axiom,
    ! [A: $tType,F4: filter(A),F5: filter(A)] :
      ( ( F4 = F5 )
    <=> ! [P8: fun(A,bool)] :
          ( eventually(A,P8,F4)
        <=> eventually(A,P8,F5) ) ) ).

% filter_eq_iff
tff(fact_7554_eventually__mono,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F4)
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => pp(aa(A,bool,Q,X3)) )
       => eventually(A,Q,F4) ) ) ).

% eventually_mono
tff(fact_7555_not__eventuallyD,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A)] :
      ( ~ eventually(A,P,F4)
     => ? [X3: A] : ~ pp(aa(A,bool,P,X3)) ) ).

% not_eventuallyD
tff(fact_7556_always__eventually,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A)] :
      ( ! [X_1: A] : pp(aa(A,bool,P,X_1))
     => eventually(A,P,F4) ) ).

% always_eventually
tff(fact_7557_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_aov(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_7558_eventually__principal,axiom,
    ! [A: $tType,P: fun(A,bool),S: set(A)] :
      ( eventually(A,P,principal(A,S))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S))
         => pp(aa(A,bool,P,X4)) ) ) ).

% eventually_principal
tff(fact_7559_eventually__Sup,axiom,
    ! [A: $tType,P: fun(A,bool),S: set(filter(A))] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S))
    <=> ! [X4: filter(A)] :
          ( pp(aa(set(filter(A)),bool,member(filter(A),X4),S))
         => eventually(A,P,X4) ) ) ).

% eventually_Sup
tff(fact_7560_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aow(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P),Net)
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X4),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_7561_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X3),Net) )
       => eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aow(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_7562_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,I2: A,I5: set(A),P: fun(B,bool),F4: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,member(A,I2),I5))
     => ( eventually(B,P,aa(A,filter(B),F4,I2))
       => eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I5))) ) ) ).

% eventually_INF1
tff(fact_7563_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ae(A,bool),F4) ) ).

% trivial_limit_def
tff(fact_7564_eventually__const__iff,axiom,
    ! [A: $tType,P: bool,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_oz(bool,fun(A,bool),P),F4)
    <=> ( pp(P)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_7565_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,bool),Net: filter(A)] :
      ( ! [X3: A] : ~ pp(aa(A,bool,P,X3))
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_7566_eventually__happens_H,axiom,
    ! [A: $tType,F4: filter(A),P: fun(A,bool)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,P,F4)
       => ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ) ).

% eventually_happens'
tff(fact_7567_eventually__happens,axiom,
    ! [A: $tType,P: fun(A,bool),Net: filter(A)] :
      ( eventually(A,P,Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ) ).

% eventually_happens
tff(fact_7568_eventually__bot,axiom,
    ! [A: $tType,P: fun(A,bool)] : eventually(A,P,bot_bot(filter(A))) ).

% eventually_bot
tff(fact_7569_eventually__inf__principal,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),S2: set(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,S2)))
    <=> eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_aox(fun(A,bool),fun(set(A),fun(A,bool)),P),S2),F4) ) ).

% eventually_inf_principal
tff(fact_7570_eventually__inf,axiom,
    ! [A: $tType,P: fun(A,bool),F4: filter(A),F5: filter(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),F5))
    <=> ? [Q9: fun(A,bool),R10: fun(A,bool)] :
          ( eventually(A,Q9,F4)
          & eventually(A,R10,F5)
          & ! [X4: A] :
              ( ( pp(aa(A,bool,Q9,X4))
                & pp(aa(A,bool,R10,X4)) )
             => pp(aa(A,bool,P,X4)) ) ) ) ).

% eventually_inf
tff(fact_7571_wo__rel_Omax2__among,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => pp(aa(set(A),bool,member(A,bNF_We1388413361240627857o_max2(A,R3,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))) ) ) ) ).

% wo_rel.max2_among
tff(fact_7572_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_aoy(fun(A,bool),fun(A,bool),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_7573_eventually__Inf__base,axiom,
    ! [A: $tType,B4: set(filter(A)),P: fun(A,bool)] :
      ( ( B4 != bot_bot(set(filter(A))) )
     => ( ! [F8: filter(A)] :
            ( pp(aa(set(filter(A)),bool,member(filter(A),F8),B4))
           => ! [G5: filter(A)] :
                ( pp(aa(set(filter(A)),bool,member(filter(A),G5),B4))
               => ? [X5: filter(A)] :
                    ( pp(aa(set(filter(A)),bool,member(filter(A),X5),B4))
                    & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),X5),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F8),G5))) ) ) )
       => ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B4))
        <=> ? [X4: filter(A)] :
              ( pp(aa(set(filter(A)),bool,member(filter(A),X4),B4))
              & eventually(A,P,X4) ) ) ) ) ).

% eventually_Inf_base
tff(fact_7574_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_aoz(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_top
tff(fact_7575_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),Z9))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_aoz(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_7576_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),F4: filter(B),G3: fun(B,A)] :
          ( filterlim(B,A,F3,at_top(A),F4)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_apa(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G3),F4)
           => filterlim(B,A,G3,at_top(A),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_7577_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_apb(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_7578_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P: fun(B,bool),F4: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),A5)))
      <=> ? [Q9: fun(A,fun(B,bool))] :
            ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),A5))
               => eventually(B,aa(A,fun(B,bool),Q9,X4),aa(A,filter(B),F4,X4)) )
            & ! [Y5: B] :
                ( ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A5))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q9,X4),Y5)) )
               => pp(aa(B,bool,P,Y5)) ) ) ) ) ).

% eventually_INF_finite
tff(fact_7579_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Z9),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_apc(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_7580_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_apc(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_bot
tff(fact_7581_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_apd(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_7582_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3))
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_7583_natLeq__on__wo__rel,axiom,
    ! [N2: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ajh(nat,fun(nat,fun(nat,bool)),N2)))) ).

% natLeq_on_wo_rel
tff(fact_7584_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3))
            & pp(aa(set(A),bool,member(A,bNF_We1388413361240627857o_max2(A,R3,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_7585_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C2),Z9))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ape(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7586_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F4: fun(B,filter(A)),B4: set(B)] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),B4)))
    <=> ? [X14: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X14),B4))
          & pp(aa(set(B),bool,finite_finite2(B),X14))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),X14))) ) ) ).

% eventually_INF
tff(fact_7587_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z9),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_apf(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_7588_eventually__Inf,axiom,
    ! [A: $tType,P: fun(A,bool),B4: set(filter(A))] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B4))
    <=> ? [X14: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X14),B4))
          & pp(aa(set(filter(A)),bool,finite_finite2(filter(A)),X14))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X14)) ) ) ).

% eventually_Inf
tff(fact_7589_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,set(B)),A5: set(B),F4: filter(A)] :
      ( filterlim(A,set(B),F3,finite5375528669736107172at_top(B,A5),F4)
    <=> ! [X14: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),X14))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X14),A5)) )
         => eventually(A,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_apg(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),F3),A5),X14),F4) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_7590_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G3: fun(B,A),Y6: set(B),X6: set(A),F4: filter(B),F3: fun(A,C)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G3),Y6)),X6))
     => ( eventually(B,aTP_Lamp_anv(set(B),fun(B,bool),Y6),F4)
       => ( map_filter_on(A,C,X6,F3,map_filter_on(B,A,Y6,G3,F4)) = map_filter_on(B,C,Y6,aa(fun(B,A),fun(B,C),comp(A,C,B,F3),G3),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_7591_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( B4 != bot_bot(set(A)) )
         => ? [X_1: A] : pp(aa(A,bool,bNF_We4791949203932849705sMinim(A,R3,B4),X_1)) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_7592_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F4: filter(A),P: fun(B,bool),F3: fun(A,B)] :
      ( eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6),F4)
     => ( eventually(B,P,map_filter_on(A,B,X6,F3,F4))
      <=> eventually(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),aTP_Lamp_aph(set(A),fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),X6),P),F3),F4) ) ) ).

% eventually_map_filter_on
tff(fact_7593_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(A,bool,bNF_We4791949203932849705sMinim(A,R3,A5),B2))
      <=> ( pp(aa(set(A),bool,member(A,B2),A5))
          & ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X4)),R3)) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_7594_wo__rel_Ominim__isMinim,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( B4 != bot_bot(set(A)) )
         => pp(aa(A,bool,bNF_We4791949203932849705sMinim(A,R3,B4),bNF_We6954850376910717587_minim(A,R3,B4))) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_7595_wo__rel_Ominim__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( bNF_We6954850376910717587_minim(A,R3,A5) = the(A,bNF_We4791949203932849705sMinim(A,R3,A5)) ) ) ).

% wo_rel.minim_def
tff(fact_7596_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,A3),B4))
         => ( ! [B5: A] :
                ( pp(aa(set(A),bool,member(A,B5),B4))
               => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B5)),R3)) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R3,B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_7597_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),B4))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R3,B4)),B2)),R3)) ) ) ) ).

% wo_rel.minim_least
tff(fact_7598_wo__rel_Ominim__inField,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( B4 != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_We6954850376910717587_minim(A,R3,B4)),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ) ) ).

% wo_rel.minim_inField
tff(fact_7599_wo__rel_Ominim__in,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( B4 != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_We6954850376910717587_minim(A,R3,B4)),B4)) ) ) ) ).

% wo_rel.minim_in
tff(fact_7600_Sup__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S) = abs_filter(A,aTP_Lamp_api(set(filter(A)),fun(fun(A,bool),bool),S)) ).

% Sup_filter_def
tff(fact_7601_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),linord329482645794927042rt_key(B,A,F3,X,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_7602_insort__insert__triv,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_rm(A,A),X,Xs) = Xs ) ) ) ).

% insort_insert_triv
tff(fact_7603_bot__filter__def,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = abs_filter(A,aTP_Lamp_apj(fun(A,bool),bool)) ).

% bot_filter_def
tff(fact_7604_sup__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F5: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F5) = abs_filter(A,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_apk(filter(A),fun(filter(A),fun(fun(A,bool),bool)),F4),F5)) ).

% sup_filter_def
tff(fact_7605_principal__def,axiom,
    ! [A: $tType,S: set(A)] : principal(A,S) = abs_filter(A,aa(set(A),fun(fun(A,bool),bool),ball(A),S)) ).

% principal_def
tff(fact_7606_map__filter__on__def,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F3: fun(A,B),F4: filter(A)] : map_filter_on(A,B,X6,F3,F4) = abs_filter(B,aa(filter(A),fun(fun(B,bool),bool),aa(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),aTP_Lamp_apm(set(A),fun(fun(A,B),fun(filter(A),fun(fun(B,bool),bool))),X6),F3),F4)) ).

% map_filter_on_def
tff(fact_7607_top__filter__def,axiom,
    ! [A: $tType] : top_top(filter(A)) = abs_filter(A,fAll(A)) ).

% top_filter_def
tff(fact_7608_set__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_rm(A,A),X,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_insert
tff(fact_7609_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),X: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_rm(A,A),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_7610_insort__insert__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_rm(A,A),X,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_rm(A,A)),X),Xs) ) ) ) ).

% insort_insert_insort
tff(fact_7611_inf__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F5: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),F5) = abs_filter(A,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_apn(filter(A),fun(filter(A),fun(fun(A,bool),bool)),F4),F5)) ).

% inf_filter_def
tff(fact_7612_Ref__Time_Oupdate__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),V2: A] : ref_update(A,R3,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_apo(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),R3),V2)) ) ).

% Ref_Time.update_def
tff(fact_7613_prod__mset_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: multiset(B),B4: multiset(B),G3: fun(B,A)] :
          ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A5),B4) = zero_zero(multiset(B)) )
         => ( comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A5),B4))) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),A5))),comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),B4))) ) ) ) ).

% prod_mset.union_disjoint
tff(fact_7614_unit__abs__eta__conv,axiom,
    ! [A: $tType,F3: fun(product_unit,A)] : aTP_Lamp_app(fun(product_unit,A),fun(product_unit,A),F3) = F3 ).

% unit_abs_eta_conv
tff(fact_7615_prod__mset_Oadd__mset,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [X: A,N7: multiset(A)] : comm_m9189036328036947845d_mset(A,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),N7)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),comm_m9189036328036947845d_mset(A,N7)) ) ).

% prod_mset.add_mset
tff(fact_7616_prod__mset_Ounion,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M6: multiset(A),N7: multiset(A)] : comm_m9189036328036947845d_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_m9189036328036947845d_mset(A,M6)),comm_m9189036328036947845d_mset(A,N7)) ) ).

% prod_mset.union
tff(fact_7617_prod__mset__Un,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: multiset(A),B4: multiset(A)] : comm_m9189036328036947845d_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_m9189036328036947845d_mset(A,A5)),comm_m9189036328036947845d_mset(A,B4)) ) ).

% prod_mset_Un
tff(fact_7618_prod__mset_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: multiset(B)] : comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_eg(B,A)),A5)) = one_one(A) ) ).

% prod_mset.neutral_const
tff(fact_7619_prod__mset_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),X: B,A5: multiset(B)] : comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),A5))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),A5))) ) ).

% prod_mset.insert
tff(fact_7620_prod__mset__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,A5: multiset(B)] : comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_ks(A,fun(B,A),C2)),A5)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(multiset(B),nat,size_size(multiset(B)),A5)) ) ).

% prod_mset_constant
tff(fact_7621_UNIV__unit,axiom,
    top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),aa(product_unit,fun(set(product_unit),set(product_unit)),insert(product_unit),product_Unity),bot_bot(set(product_unit))) ).

% UNIV_unit
tff(fact_7622_bot__unit__def,axiom,
    bot_bot(product_unit) = product_Unity ).

% bot_unit_def
tff(fact_7623_Unity__def,axiom,
    product_Unity = aa(bool,product_unit,product_Abs_unit,fTrue) ).

% Unity_def
tff(fact_7624_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),sup_sup(product_unit),Uu),Uv) = product_Unity ).

% sup_unit_def
tff(fact_7625_Inf__unit__def,axiom,
    ! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Inf_Inf(product_unit),Uu) = product_Unity ).

% Inf_unit_def
tff(fact_7626_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : Y = product_Unity ).

% old.unit.exhaust
tff(fact_7627_top__unit__def,axiom,
    top_top(product_unit) = product_Unity ).

% top_unit_def
tff(fact_7628_uminus__unit__def,axiom,
    ! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).

% uminus_unit_def
tff(fact_7629_prod__mset_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),H: fun(B,A),A5: multiset(B)] : comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_en(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A5)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),A5))),comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,H),A5))) ) ).

% prod_mset.distrib
tff(fact_7630_prod__mset_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,fun(C,A)),B4: multiset(C),A5: multiset(B)] : comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_apq(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),G3),B4)),A5)) = comm_m9189036328036947845d_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_apr(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),G3),A5)),B4)) ) ).

% prod_mset.swap
tff(fact_7631_Sup__unit__def,axiom,
    ! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Sup_Sup(product_unit),Uu) = product_Unity ).

% Sup_unit_def
tff(fact_7632_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),inf_inf(product_unit),Uu),Uv) = product_Unity ).

% inf_unit_def
tff(fact_7633_wait__def,axiom,
    ! [N2: nat] : heap_Time_wait(N2) = heap_Time_Heap2(product_unit,aTP_Lamp_aps(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N2)) ).

% wait_def
tff(fact_7634_prod__mset_Oeq__fold,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M6: multiset(A)] : comm_m9189036328036947845d_mset(A,M6) = fold_mset(A,A,times_times(A),one_one(A),M6) ) ).

% prod_mset.eq_fold
tff(fact_7635_prod__mset__delta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: B,C2: A,A5: multiset(B)] : comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_apt(B,fun(A,fun(B,A)),Y),C2)),A5)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A5),Y)) ) ).

% prod_mset_delta'
tff(fact_7636_prod__mset__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: B,C2: A,A5: multiset(B)] : comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_apu(B,fun(A,fun(B,A)),Y),C2)),A5)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A5),Y)) ) ).

% prod_mset_delta
tff(fact_7637_prod__mset__multiplicity,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M6: multiset(A)] : comm_m9189036328036947845d_mset(A,M6) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_apv(multiset(A),fun(A,A),M6)),aa(multiset(A),set(A),set_mset(A),M6)) ) ).

% prod_mset_multiplicity
tff(fact_7638_execute__update,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),V2: A,H: 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,R3,V2)),H) = 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,R3,V2,H)),one_one(nat)))) ) ).

% execute_update
tff(fact_7639_prod__mset_Oremove,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [X: A,A5: multiset(A)] :
          ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),A5)))
         => ( comm_m9189036328036947845d_mset(A,A5) = aa(A,A,aa(A,fun(A,A),times_times(A),X),comm_m9189036328036947845d_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))))) ) ) ) ).

% prod_mset.remove
tff(fact_7640_old_Ounit_Orec,axiom,
    ! [T: $tType,F1: T] : product_rec_unit(T,F1,product_Unity) = F1 ).

% old.unit.rec
tff(fact_7641_default__unit__def,axiom,
    default_default(product_unit) = product_Unity ).

% default_unit_def
tff(fact_7642_old_Orec__unit__def,axiom,
    ! [T: $tType,X5: T,Xa3: product_unit] : product_rec_unit(T,X5,Xa3) = the(T,product_rec_set_unit(T,X5,Xa3)) ).

% old.rec_unit_def
tff(fact_7643_CODE__ABORT__def,axiom,
    ! [A: $tType,F3: fun(product_unit,A)] : cODE_ABORT(A,F3) = aa(product_unit,A,F3,product_Unity) ).

% CODE_ABORT_def
tff(fact_7644_log_Osimps,axiom,
    ! [B2: code_natural,I2: code_natural] :
      ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
          | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),B2)) )
       => ( log(B2,I2) = one_one(code_natural) ) )
      & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
            | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),B2)) )
       => ( log(B2,I2) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(B2,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),I2),B2))) ) ) ) ).

% log.simps
tff(fact_7645_log_Oelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
            | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
         => ( Y = one_one(code_natural) ) )
        & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
              | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
         => ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa2),X))) ) ) ) ) ).

% log.elims
tff(fact_7646_log_Opelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( pp(aa(product_prod(code_natural,code_natural),bool,accp(product_prod(code_natural,code_natural),log_rel),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X),Xa2)))
       => ~ ( ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
                  | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
               => ( Y = one_one(code_natural) ) )
              & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
                    | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
               => ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa2),X))) ) ) )
           => ~ pp(aa(product_prod(code_natural,code_natural),bool,accp(product_prod(code_natural,code_natural),log_rel),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X),Xa2))) ) ) ) ).

% log.pelims
tff(fact_7647_minus__shift__def,axiom,
    ! [K: code_natural,L: code_natural,R3: code_natural] :
      ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K),L))
       => ( minus_shift(R3,K,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),R3),K)),L) ) )
      & ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K),L))
       => ( minus_shift(R3,K,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),L) ) ) ) ).

% minus_shift_def
tff(fact_7648_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))] :
      ~ ! [F2: fun(code_natural,A),N5: code_natural,M5: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N5),M5)) ).

% Lazy_Sequence.iterate_upto.cases
tff(fact_7649_exhaustive__fun_H_Ocases,axiom,
    ! [A: $tType,B: $tType] :
      ( ( quickc658316121487927005ustive(B)
        & cl_HOL_Oequal(A)
        & quickc658316121487927005ustive(A) )
     => ! [X: product_prod(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
          ~ ! [F2: fun(fun(A,B),option(product_prod(bool,list(code_term)))),I3: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(fun(A,B),option(product_prod(bool,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),F2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I3),D2)) ) ).

% exhaustive_fun'.cases
tff(fact_7650_exhaustive__natural_H_Ocases,axiom,
    ! [X: product_prod(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
      ~ ! [F2: fun(code_natural,option(product_prod(bool,list(code_term)))),D2: code_natural,I3: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(code_natural,option(product_prod(bool,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),F2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),D2),I3)) ).

% exhaustive_natural'.cases
tff(fact_7651_full__exhaustive__fun_H_Ocases,axiom,
    ! [A: $tType,B: $tType] :
      ( ( quickc3360725361186068524ustive(B)
        & cl_HOL_Oequal(A)
        & quickc3360725361186068524ustive(A) )
     => ! [X: product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
          ~ ! [F2: fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),I3: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,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(bool,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(bool,list(code_term)))),product_prod(code_natural,code_natural)),F2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I3),D2)) ) ).

% full_exhaustive_fun'.cases
tff(fact_7652_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
      ~ ! [F2: fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: code_natural,I3: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,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(bool,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),F2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),D2),I3)) ).

% full_exhaustive_natural'.cases
tff(fact_7653_log_Ocases,axiom,
    ! [X: product_prod(code_natural,code_natural)] :
      ~ ! [B5: code_natural,I3: code_natural] : X != aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),B5),I3) ).

% log.cases
tff(fact_7654_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)) = aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2)))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))) ).

% next.simps
tff(fact_7655_abstract__filter__def,axiom,
    ! [A: $tType,F3: fun(product_unit,filter(A))] : abstract_filter(A,F3) = aa(product_unit,filter(A),F3,product_Unity) ).

% abstract_filter_def
tff(fact_7656_iter_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))] :
      ~ ! [Random: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),Nrandom: code_natural,Seed: product_prod(code_natural,code_natural)] : X != aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),aa(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),fun(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))),product_Pair(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),Random),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),Nrandom),Seed)) ).

% iter.cases
tff(fact_7657_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_apx(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_7658_Random_Orange__def,axiom,
    ! [K: code_natural] : range(K) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),K),aTP_Lamp_apz(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_aqa(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K)) ).

% Random.range_def
tff(fact_7659_scomp__apply,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(B,product_prod(C,D)),G3: fun(C,fun(D,A)),X: B] : aa(B,A,product_scomp(B,C,D,A,F3,G3),X) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),G3),aa(B,product_prod(C,D),F3,X)) ).

% scomp_apply
tff(fact_7660_Pair__scomp,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: C,F3: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X),F3) = aa(C,fun(A,B),F3,X) ).

% Pair_scomp
tff(fact_7661_scomp__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,product_prod(B,C))] : product_scomp(A,B,C,product_prod(B,C),X,product_Pair(B,C)) = X ).

% scomp_Pair
tff(fact_7662_scomp__scomp,axiom,
    ! [A: $tType,E: $tType,F: $tType,B: $tType,D: $tType,C: $tType,F3: fun(A,product_prod(E,F)),G3: fun(E,fun(F,product_prod(C,D))),H: fun(C,fun(D,B))] : product_scomp(A,C,D,B,product_scomp(A,E,F,product_prod(C,D),F3,G3),H) = product_scomp(A,E,F,B,F3,aa(fun(C,fun(D,B)),fun(E,fun(F,B)),aTP_Lamp_aqb(fun(E,fun(F,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F,B))),G3),H)) ).

% scomp_scomp
tff(fact_7663_scomp__def,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,F3: fun(A,product_prod(B,C)),G3: fun(B,fun(C,D)),X5: A] : aa(A,D,product_scomp(A,B,C,D,F3,G3),X5) = aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G3),aa(A,product_prod(B,C),F3,X5)) ).

% scomp_def
tff(fact_7664_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa2),Xb) = Y )
     => ( ( ( X = zero_zero(code_natural) )
         => ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
        & ( ( X != zero_zero(code_natural) )
         => ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa2)) ) ) ) ) ).

% iterate.elims
tff(fact_7665_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K: code_natural,F3: fun(B,fun(A,product_prod(B,A))),X: B] :
      ( ( ( K = zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F3),X) = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X) ) )
      & ( ( K != zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F3),X) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F3,X),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F3)) ) ) ) ).

% iterate.simps
tff(fact_7666_scomp__unfold,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,X5: fun(A,product_prod(B,C)),Xa3: fun(B,fun(C,D)),Xb2: A] : aa(A,D,product_scomp(A,B,C,D,X5,Xa3),Xb2) = aa(C,D,aa(B,fun(C,D),Xa3,aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),X5,Xb2))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X5,Xb2))) ).

% scomp_unfold
tff(fact_7667_range,axiom,
    ! [K: code_natural,S2: product_prod(code_natural,code_natural)] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),K))
     => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),code_natural,product_fst(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)),range(K),S2))),K)) ) ).

% range
tff(fact_7668_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa2),Xb) = Y )
     => ( pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb))))
       => ~ ( ( ( ( X = zero_zero(code_natural) )
               => ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
              & ( ( X != zero_zero(code_natural) )
               => ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa2)) ) ) )
           => ~ pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb)))) ) ) ) ).

% iterate.pelims
tff(fact_7669_select__weight__member,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A)),S2: product_prod(code_natural,code_natural)] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),groups8242544230860333062m_list(code_natural,aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))))
     => pp(aa(set(A),bool,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,Xs),S2))),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),map(product_prod(code_natural,A),A,product_snd(code_natural,A)),Xs)))) ) ).

% select_weight_member
tff(fact_7670_iterate_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))] :
      ~ ! [K2: code_natural,F2: fun(B,fun(A,product_prod(B,A))),X3: B] : X != 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)),K2),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),F2),X3)) ).

% iterate.cases
tff(fact_7671_select__weight__cons__zero,axiom,
    ! [A: $tType,X: A,Xs: list(product_prod(code_natural,A))] : select_weight(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),aa(product_prod(code_natural,A),fun(list(product_prod(code_natural,A)),list(product_prod(code_natural,A))),cons(product_prod(code_natural,A)),aa(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),zero_zero(code_natural)),X)),Xs)) = select_weight(A,Xs) ).

% select_weight_cons_zero
tff(fact_7672_select__weight__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,bool)),fun(product_prod(code_natural,A),bool),product_case_prod(code_natural,A,bool),aTP_Lamp_aqc(code_natural,fun(A,bool)))),Xs)) = select_weight(A,Xs) ).

% select_weight_drop_zero
tff(fact_7673_select__weight__def,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(groups8242544230860333062m_list(code_natural,aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))),aTP_Lamp_aqd(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).

% select_weight_def
tff(fact_7674_pick__member,axiom,
    ! [A: $tType,I2: code_natural,Xs: list(product_prod(code_natural,A))] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),groups8242544230860333062m_list(code_natural,aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))))
     => pp(aa(set(A),bool,member(A,aa(code_natural,A,pick(A,Xs),I2)),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),map(product_prod(code_natural,A),A,product_snd(code_natural,A)),Xs)))) ) ).

% pick_member
tff(fact_7675_pick__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : pick(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,bool)),fun(product_prod(code_natural,A),bool),product_case_prod(code_natural,A,bool),aTP_Lamp_aqc(code_natural,fun(A,bool)))),Xs)) = pick(A,Xs) ).

% pick_drop_zero
tff(fact_7676_pick_Osimps,axiom,
    ! [A: $tType,I2: code_natural,X: product_prod(code_natural,A),Xs: list(product_prod(code_natural,A))] :
      ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
       => ( aa(code_natural,A,pick(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)),X),Xs)),I2) = aa(product_prod(code_natural,A),A,product_snd(code_natural,A),X) ) )
      & ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
       => ( aa(code_natural,A,pick(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)),X),Xs)),I2) = aa(code_natural,A,pick(A,Xs),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),I2),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X))) ) ) ) ).

% pick.simps
tff(fact_7677_select__weight__select,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( select_weight(A,aa(list(A),list(product_prod(code_natural,A)),map(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),one_one(code_natural))),Xs)) = select(A,Xs) ) ) ).

% select_weight_select
tff(fact_7678_Predicate_Oiterate__upto_Opinduct,axiom,
    ! [A: $tType,A0: fun(code_natural,A),A1: code_natural,A22: code_natural,P: fun(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)))] :
      ( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),A0),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),A1),A22))))
     => ( ! [F2: fun(code_natural,A),N5: code_natural,M5: code_natural] :
            ( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,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),N5),M5))))
           => ( ! [X5: product_unit] :
                  ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),M5),N5))
                 => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,F2),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N5),one_one(code_natural))),M5)) )
             => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,F2),N5),M5)) ) )
       => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,A0),A1),A22)) ) ) ).

% Predicate.iterate_upto.pinduct
tff(fact_7679_pick__same,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(code_natural,A,pick(A,aa(list(A),list(product_prod(code_natural,A)),map(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),one_one(code_natural))),Xs)),code_natural_of_nat(L)) = aa(nat,A,nth(A,Xs),L) ) ) ).

% pick_same
tff(fact_7680_filtercomap__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F4: filter(B)] : filtercomap(A,B,F3,F4) = abs_filter(A,aa(filter(B),fun(fun(A,bool),bool),aTP_Lamp_aqe(fun(A,B),fun(filter(B),fun(fun(A,bool),bool)),F3),F4)) ).

% filtercomap_def
tff(fact_7681_filterlim__filtercomap,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F4: filter(B)] : filterlim(A,B,F3,F4,filtercomap(A,B,F3,F4)) ).

% filterlim_filtercomap
tff(fact_7682_filtercomap__bot,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : filtercomap(A,B,F3,bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtercomap_bot
tff(fact_7683_eventually__filtercomapI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F4: filter(A),F3: fun(B,A)] :
      ( eventually(A,P,F4)
     => eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aol(fun(A,bool),fun(fun(B,A),fun(B,bool)),P),F3),filtercomap(B,A,F3,F4)) ) ).

% eventually_filtercomapI
tff(fact_7684_filtercomap__top,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : filtercomap(A,B,F3,top_top(filter(B))) = top_top(filter(A)) ).

% filtercomap_top
tff(fact_7685_filtercomap__principal,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] : filtercomap(A,B,F3,principal(B,A5)) = principal(A,aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A5)) ).

% filtercomap_principal
tff(fact_7686_less__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),code_natural_of_nat(Xa2)),code_natural_of_nat(X)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),X)) ) ).

% less_natural.abs_eq
tff(fact_7687_filtercomap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),F4: fun(C,filter(B)),B4: set(C)] : filtercomap(A,B,F3,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B4))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aqf(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F3),F4)),B4)) ).

% filtercomap_INF
tff(fact_7688_filtercomap__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filtercomap(A,A,aTP_Lamp_cq(A,A),F4) = F4 ).

% filtercomap_ident
tff(fact_7689_filtercomap__filtercomap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),G3: fun(B,C),F4: filter(C)] : filtercomap(A,B,F3,filtercomap(B,C,G3,F4)) = filtercomap(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqg(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),F4) ).

% filtercomap_filtercomap
tff(fact_7690_filterlim__filtercomap__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,B),G3: fun(B,C),G4: filter(C),F4: filter(A)] :
      ( filterlim(A,B,F3,filtercomap(B,C,G3,G4),F4)
    <=> filterlim(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G3),F3),G4,F4) ) ).

% filterlim_filtercomap_iff
tff(fact_7691_eventually__filtercomap,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F3: fun(A,B),F4: filter(B)] :
      ( eventually(A,P,filtercomap(A,B,F3,F4))
    <=> ? [Q9: fun(B,bool)] :
          ( eventually(B,Q9,F4)
          & ! [X4: A] :
              ( pp(aa(B,bool,Q9,aa(A,B,F3,X4)))
             => pp(aa(A,bool,P,X4)) ) ) ) ).

% eventually_filtercomap
tff(fact_7692_filtercomap__neq__bot,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F3: fun(B,A)] :
      ( ! [P2: fun(A,bool)] :
          ( eventually(A,P2,F4)
         => ? [X5: B] : pp(aa(A,bool,P2,aa(B,A,F3,X5))) )
     => ( filtercomap(B,A,F3,F4) != bot_bot(filter(B)) ) ) ).

% filtercomap_neq_bot
tff(fact_7693_filtercomap__inf,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F14: filter(B),F23: filter(B)] : filtercomap(A,B,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F14),F23)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),filtercomap(A,B,F3,F14)),filtercomap(A,B,F3,F23)) ).

% filtercomap_inf
tff(fact_7694_filtercomap__sup,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F14: filter(B),F23: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),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,F3,F14)),filtercomap(A,B,F3,F23))),filtercomap(A,B,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F14),F23)))) ).

% filtercomap_sup
tff(fact_7695_filterlim__iff__le__filtercomap,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F4: filter(B),G4: filter(A)] :
      ( filterlim(A,B,F3,F4,G4)
    <=> pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G4),filtercomap(A,B,F3,F4))) ) ).

% filterlim_iff_le_filtercomap
tff(fact_7696_filtercomap__mono,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),F5: filter(A),F3: fun(B,A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F5))
     => pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtercomap(B,A,F3,F4)),filtercomap(B,A,F3,F5))) ) ).

% filtercomap_mono
tff(fact_7697_less__eq__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),code_natural_of_nat(Xa2)),code_natural_of_nat(X)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X)) ) ).

% less_eq_natural.abs_eq
tff(fact_7698_filtercomap__neq__bot__surj,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F3: fun(B,A)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
       => ( filtercomap(B,A,F3,F4) != bot_bot(filter(B)) ) ) ) ).

% filtercomap_neq_bot_surj
tff(fact_7699_eventually__filtercomap__at__top__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F3,at_top(A)))
        <=> ? [N12: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N12),aa(B,A,F3,X4)))
             => pp(aa(B,bool,P,X4)) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_7700_eventually__filtercomap__at__top__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F3,at_top(A)))
        <=> ? [N12: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N12),aa(B,A,F3,X4)))
             => pp(aa(B,bool,P,X4)) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_7701_eventually__filtercomap__at__bot__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F3,at_bot(A)))
        <=> ? [N12: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),N12))
             => pp(aa(B,bool,P,X4)) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_7702_eventually__filtercomap__at__bot__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F3,at_bot(A)))
        <=> ? [N12: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),N12))
             => pp(aa(B,bool,P,X4)) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_7703_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,C),F4: fun(B,filter(C)),B4: set(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_aqh(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F3),F4)),B4))),filtercomap(A,C,F3,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F4),B4))))) ).

% filtercomap_SUP
tff(fact_7704_select__def,axiom,
    ! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(code_natural_of_nat(aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_aqi(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).

% select_def
tff(fact_7705_iter_H_Ocases,axiom,
    ! [A: $tType] :
      ( quickcheck_random(A)
     => ! [X: product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))] :
          ~ ! [T7: itself(A),Nrandom: code_natural,Sz: code_natural,Seed: product_prod(code_natural,code_natural)] : X != aa(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),aa(itself(A),fun(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))),product_Pair(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),T7),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_7706_less__eq__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),Xa2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2))) ) ).

% less_eq_natural.rep_eq
tff(fact_7707_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),X),Xa2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2))) ) ).

% less_natural.rep_eq
tff(fact_7708_less__natural__def,axiom,
    ord_less(code_natural) = aa(fun(nat,fun(nat,bool)),fun(code_natural,fun(code_natural,bool)),map_fun(code_natural,nat,fun(nat,bool),fun(code_natural,bool),code_nat_of_natural,map_fun(code_natural,nat,bool,bool,code_nat_of_natural,id(bool))),ord_less(nat)) ).

% less_natural_def
tff(fact_7709_less__eq__natural__def,axiom,
    ord_less_eq(code_natural) = aa(fun(nat,fun(nat,bool)),fun(code_natural,fun(code_natural,bool)),map_fun(code_natural,nat,fun(nat,bool),fun(code_natural,bool),code_nat_of_natural,map_fun(code_natural,nat,bool,bool,code_nat_of_natural,id(bool))),ord_less_eq(nat)) ).

% less_eq_natural_def
tff(fact_7710_natural__decr,axiom,
    ! [N2: code_natural] :
      ( ( N2 != zero_zero(code_natural) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,N2)),aa(nat,nat,suc,zero_zero(nat)))),aa(code_natural,nat,code_nat_of_natural,N2))) ) ).

% natural_decr
tff(fact_7711_admissible__chfin,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,bool)] :
          ( ! [S8: set(A)] :
              ( comple1602240252501008431_chain(A,ord_less_eq(A),S8)
             => pp(aa(set(A),bool,finite_finite2(A),S8)) )
         => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P) ) ) ).

% admissible_chfin
tff(fact_7712_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_concat_bit(M,K,L)),N2))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) )
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_7713_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_concat_bit(N2,K,L)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_7714_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_concat_bit(N2,K,L)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ).

% concat_bit_negative_iff
tff(fact_7715_admissible__ball,axiom,
    ! [A: $tType,B: $tType,A5: set(A),Lub: fun(set(B),B),Ord: fun(B,fun(B,bool)),P: fun(B,fun(A,bool))] :
      ( ! [Y3: A] :
          ( pp(aa(set(A),bool,member(A,Y3),A5))
         => comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),Y3)) )
     => comple1908693960933563346ssible(B,Lub,Ord,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aow(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P)) ) ).

% admissible_ball
tff(fact_7716_admissible__const,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),T6: bool] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_oz(bool,fun(A,bool),T6)) ).

% admissible_const
tff(fact_7717_admissible__conj,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P: fun(A,bool),Q: fun(A,bool)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
     => ( comple1908693960933563346ssible(A,Lub,Ord,Q)
       => comple1908693960933563346ssible(A,Lub,Ord,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ag(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) ) ) ).

% admissible_conj
tff(fact_7718_admissible__True,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool))] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_bm(A,bool)) ).

% admissible_True
tff(fact_7719_admissible__all,axiom,
    ! [A: $tType,B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,bool)),P: fun(B,fun(A,bool))] :
      ( ! [Y3: A] : comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),Y3))
     => comple1908693960933563346ssible(B,Lub,Ord,aTP_Lamp_aqj(fun(B,fun(A,bool)),fun(B,bool),P)) ) ).

% admissible_all
tff(fact_7720_ccpo_Oadmissible__def,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P: fun(A,bool)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
    <=> ! [A13: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A13)
         => ( ( A13 != bot_bot(set(A)) )
           => ( ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A13))
                 => pp(aa(A,bool,P,X4)) )
             => pp(aa(A,bool,P,aa(set(A),A,Lub,A13))) ) ) ) ) ).

% ccpo.admissible_def
tff(fact_7721_ccpo_OadmissibleI,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),P: fun(A,bool),Lub: fun(set(A),A)] :
      ( ! [A11: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A11)
         => ( ( A11 != bot_bot(set(A)) )
           => ( ! [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),A11))
                 => pp(aa(A,bool,P,X5)) )
             => pp(aa(A,bool,P,aa(set(A),A,Lub,A11))) ) ) )
     => comple1908693960933563346ssible(A,Lub,Ord,P) ) ).

% ccpo.admissibleI
tff(fact_7722_ccpo_OadmissibleD,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P: fun(A,bool),A5: set(A)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
     => ( comple1602240252501008431_chain(A,Ord,A5)
       => ( ( A5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => pp(aa(A,bool,P,X3)) )
           => pp(aa(A,bool,P,aa(set(A),A,Lub,A5))) ) ) ) ) ).

% ccpo.admissibleD
tff(fact_7723_admissible__disj,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
         => ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),Q)
           => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aqk(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) ) ) ) ).

% admissible_disj
tff(fact_7724_multp__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),M6: multiset(A),N7: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),M6),N7))
    <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,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)),M6),N7)),mult(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% multp_def
tff(fact_7725_pair__lessI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T6))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T6))),fun_pair_less)) ) ) ).

% pair_lessI2
tff(fact_7726_total__pair__less,axiom,
    ! [A5: set(product_prod(nat,nat))] : total_on(product_prod(nat,nat),A5,fun_pair_less) ).

% total_pair_less
tff(fact_7727_trans__pair__less,axiom,
    trans(product_prod(nat,nat),fun_pair_less) ).

% trans_pair_less
tff(fact_7728_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Z2))),fun_pair_less))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),Z2)) ) ).

% pair_less_iff1
tff(fact_7729_pair__less__def,axiom,
    fun_pair_less = lex_prod(nat,nat,less_than,less_than) ).

% pair_less_def
tff(fact_7730_wf__pair__less,axiom,
    wf(product_prod(nat,nat),fun_pair_less) ).

% wf_pair_less
tff(fact_7731_less__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
        <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,ord_less(A)),M6),N7)) ) ) ).

% less_multiset_def
tff(fact_7732_pair__lessI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T6))),fun_pair_less)) ) ).

% pair_lessI1
tff(fact_7733_mono__multp,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),R6: fun(A,fun(A,bool))] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R3),R6))
     => pp(aa(fun(multiset(A),fun(multiset(A),bool)),bool,aa(fun(multiset(A),fun(multiset(A),bool)),fun(fun(multiset(A),fun(multiset(A),bool)),bool),ord_less_eq(fun(multiset(A),fun(multiset(A),bool))),multp(A,R3)),multp(A,R6))) ) ).

% mono_multp
tff(fact_7734_smin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),X),XS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_strict))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),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))),insert(product_prod(nat,nat)),Y),YS))),fun_min_strict)) ) ) ) ).

% smin_insertI
tff(fact_7735_pair__leqI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S2),T6))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T6))),fun_pair_leq)) ) ) ).

% pair_leqI2
tff(fact_7736_pair__leq__def,axiom,
    fun_pair_leq = aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),sup_sup(set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),fun_pair_less),id2(product_prod(nat,nat))) ).

% pair_leq_def
tff(fact_7737_smin__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] :
      ( ( X6 != bot_bot(set(product_prod(nat,nat))) )
     => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_strict)) ) ).

% smin_emptyI
tff(fact_7738_min__strict__def,axiom,
    fun_min_strict = min_ext(product_prod(nat,nat),fun_pair_less) ).

% min_strict_def
tff(fact_7739_pair__leqI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T6))),fun_pair_leq)) ) ).

% pair_leqI1
tff(fact_7740_wmax__insertI,axiom,
    ! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),Y),YS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_max_weak))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),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))),insert(product_prod(nat,nat)),X),XS)),YS)),fun_max_weak)) ) ) ) ).

% wmax_insertI
tff(fact_7741_wmin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),X),XS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_weak))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),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))),insert(product_prod(nat,nat)),Y),YS))),fun_min_weak)) ) ) ) ).

% wmin_insertI
tff(fact_7742_wmax__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),X6))
     => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),X6)),fun_max_weak)) ) ).

% wmax_emptyI
tff(fact_7743_wmin__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] : pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_weak)) ).

% wmin_emptyI
tff(fact_7744_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_7745_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))))),insert(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_7746_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))))),insert(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_7747_smax__insertI,axiom,
    ! [Y: product_prod(nat,nat),Y6: set(product_prod(nat,nat)),X: product_prod(nat,nat),X6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),Y),Y6))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),Y6)),fun_max_strict))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),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))),insert(product_prod(nat,nat)),X),X6)),Y6)),fun_max_strict)) ) ) ) ).

% smax_insertI
tff(fact_7748_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,B2)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_7749_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),euclid6346220572633701492n_size(A,B2)))
        <=> ( B2 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_7750_max__strict__def,axiom,
    fun_max_strict = max_ext(product_prod(nat,nat),fun_pair_less) ).

% max_strict_def
tff(fact_7751_euclidean__size__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] : euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2)) ) ).

% euclidean_size_mult
tff(fact_7752_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ).

% size_mult_mono
tff(fact_7753_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))) ) ) ).

% size_mult_mono'
tff(fact_7754_euclidean__size__times__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = euclid6346220572633701492n_size(A,B2) ) ) ) ).

% euclidean_size_times_unit
tff(fact_7755_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( ( B2 != zero_zero(A) )
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_7756_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_7757_dvd__imp__size__le,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ( B2 != zero_zero(A) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ).

% dvd_imp_size_le
tff(fact_7758_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,modulo_modulo(A,A3,B2))),euclid6346220572633701492n_size(A,B2))) ) ) ).

% mod_size_less
tff(fact_7759_smax__emptyI,axiom,
    ! [Y6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),Y6))
     => ( ( Y6 != bot_bot(set(product_prod(nat,nat))) )
       => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y6)),fun_max_strict)) ) ) ).

% smax_emptyI
tff(fact_7760_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( ( B2 != zero_zero(A) )
           => ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
             => ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ) )
         => ( ( ( B2 != zero_zero(A) )
             => ! [Q5: A,R: A] :
                  ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),euclid6346220572633701492n_size(A,B2)))
                   => ( ( R != zero_zero(A) )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q5 )
                       => ( ( modulo_modulo(A,A3,B2) = R )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B2)),R) ) ) ) ) ) ) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_7761_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R3: A,Q3: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R3) = A3 )
               => ( modulo_modulo(A,A3,B2) = R3 ) ) ) ) ) ) ).

% mod_eqI
tff(fact_7762_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_7763_division__segment__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( euclid7384307370059645450egment(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),euclid7384307370059645450egment(A,B2)) ) ) ) ) ).

% division_segment_mult
tff(fact_7764_division__segment__int__def,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
       => ( euclid7384307370059645450egment(int,K) = one_one(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
       => ( euclid7384307370059645450egment(int,K) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% division_segment_int_def
tff(fact_7765_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] :
          ( ( euclid7384307370059645450egment(A,A3) = euclid7384307370059645450egment(A,B2) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2)))
              | ( B2 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_7766_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R3: A,Q3: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R3) = A3 )
               => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q3 ) ) ) ) ) ) ).

% div_eqI
tff(fact_7767_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R3: A,Q3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R3)),B2) = Q3 ) ) ) ) ) ).

% div_bounded
tff(fact_7768_wmsI,axiom,
    ! [A5: multiset(product_prod(nat,nat)),B4: multiset(product_prod(nat,nat)),Z4: multiset(product_prod(nat,nat))] :
      ( ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A5)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B4))),fun_max_strict))
        | ( ( A5 = zero_zero(multiset(product_prod(nat,nat))) )
          & ( B4 = zero_zero(multiset(product_prod(nat,nat))) ) ) )
     => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,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))),Z4),A5)),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))),Z4),B4))),ms_weak)) ) ).

% wmsI
tff(fact_7769_smsI,axiom,
    ! [A5: multiset(product_prod(nat,nat)),B4: multiset(product_prod(nat,nat)),Z4: multiset(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A5)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B4))),fun_max_strict))
     => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,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))),Z4),A5)),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))),Z4),B4))),ms_strict)) ) ).

% smsI
tff(fact_7770_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_7771_ms__weak__def,axiom,
    ms_weak = aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),fun(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),sup_sup(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),ms_strict),id2(multiset(product_prod(nat,nat)))) ).

% ms_weak_def
tff(fact_7772_ms__strictI,axiom,
    ! [Z4: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat)),A5: multiset(product_prod(nat,nat)),B4: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z4,Z10)
     => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A5)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B4))),fun_max_strict))
       => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,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))),Z4),A5)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),B4))),ms_strict)) ) ) ).

% ms_strictI
tff(fact_7773_ms__weakI1,axiom,
    ! [Z4: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat)),A5: multiset(product_prod(nat,nat)),B4: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z4,Z10)
     => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A5)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B4))),fun_max_strict))
       => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,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))),Z4),A5)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),B4))),ms_weak)) ) ) ).

% ms_weakI1
tff(fact_7774_pw__leq__lstep,axiom,
    ! [X: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
     => pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X),zero_zero(multiset(product_prod(nat,nat)))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y),zero_zero(multiset(product_prod(nat,nat))))) ) ).

% pw_leq_lstep
tff(fact_7775_pw__leq__step,axiom,
    ! [X: product_prod(nat,nat),Y: product_prod(nat,nat),X6: multiset(product_prod(nat,nat)),Y6: multiset(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
     => ( pw_leq(X6,Y6)
       => pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X),zero_zero(multiset(product_prod(nat,nat))))),X6),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y),zero_zero(multiset(product_prod(nat,nat))))),Y6)) ) ) ).

% pw_leq_step
tff(fact_7776_pw__leq_Osimps,axiom,
    ! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
      ( pw_leq(A1,A22)
    <=> ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
          & ( A22 = zero_zero(multiset(product_prod(nat,nat))) ) )
        | ? [X4: product_prod(nat,nat),Y5: product_prod(nat,nat),X14: multiset(product_prod(nat,nat)),Y9: multiset(product_prod(nat,nat))] :
            ( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X4),zero_zero(multiset(product_prod(nat,nat))))),X14) )
            & ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y5),zero_zero(multiset(product_prod(nat,nat))))),Y9) )
            & pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X4),Y5)),fun_pair_leq))
            & pw_leq(X14,Y9) ) ) ) ).

% pw_leq.simps
tff(fact_7777_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))) ) )
       => ~ ! [X3: product_prod(nat,nat),Y3: product_prod(nat,nat),X9: multiset(product_prod(nat,nat))] :
              ( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X3),zero_zero(multiset(product_prod(nat,nat))))),X9) )
             => ! [Y10: multiset(product_prod(nat,nat))] :
                  ( ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y3),zero_zero(multiset(product_prod(nat,nat))))),Y10) )
                 => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X3),Y3)),fun_pair_leq))
                   => ~ pw_leq(X9,Y10) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_7778_ms__weakI2,axiom,
    ! [Z4: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z4,Z10)
     => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,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))),Z4),zero_zero(multiset(product_prod(nat,nat))))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),zero_zero(multiset(product_prod(nat,nat)))))),ms_weak)) ) ).

% ms_weakI2
tff(fact_7779_pw__leq__split,axiom,
    ! [X6: multiset(product_prod(nat,nat)),Y6: multiset(product_prod(nat,nat))] :
      ( pw_leq(X6,Y6)
     => ? [A11: multiset(product_prod(nat,nat)),B10: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat))] :
          ( ( X6 = 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))),A11),Z8) )
          & ( Y6 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),B10),Z8) )
          & ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A11)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B10))),fun_max_strict))
            | ( ( B10 = zero_zero(multiset(product_prod(nat,nat))) )
              & ( A11 = zero_zero(multiset(product_prod(nat,nat))) ) ) ) ) ) ).

% pw_leq_split
tff(fact_7780_coinduct3__lemma,axiom,
    ! [A: $tType,X6: set(A),F3: fun(set(A),set(A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_aql(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F3)))))
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_aql(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F3))),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_aql(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F3))))) ) ) ).

% coinduct3_lemma
tff(fact_7781_def__coinduct3,axiom,
    ! [A: $tType,A5: set(A),F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( ( A5 = complete_lattice_gfp(set(A),F3) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( pp(aa(set(A),bool,member(A,A3),X6))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_aqm(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),A5),F3),X6)))))
           => pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ) ).

% def_coinduct3
tff(fact_7782_def__Collect__coinduct,axiom,
    ! [A: $tType,A5: set(A),P: fun(set(A),fun(A,bool)),A3: A,X6: set(A)] :
      ( ( A5 = complete_lattice_gfp(set(A),aTP_Lamp_aqn(fun(set(A),fun(A,bool)),fun(set(A),set(A)),P)) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),aTP_Lamp_aqn(fun(set(A),fun(A,bool)),fun(set(A),set(A)),P)))
       => ( pp(aa(set(A),bool,member(A,A3),X6))
         => ( ! [Z3: A] :
                ( pp(aa(set(A),bool,member(A,Z3),X6))
               => pp(aa(A,bool,aa(set(A),fun(A,bool),P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),A5)),Z3)) )
           => pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ) ).

% def_Collect_coinduct
tff(fact_7783_gfp__fun__UnI2,axiom,
    ! [A: $tType,F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
     => ( pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3)))
       => pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),complete_lattice_gfp(set(A),F3))))) ) ) ).

% gfp_fun_UnI2
tff(fact_7784_gfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T6: A] : complete_lattice_gfp(A,aTP_Lamp_acy(A,fun(A,A),T6)) = T6 ) ).

% gfp_const
tff(fact_7785_gfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G3: fun(A,B),F3: fun(B,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),G3))
         => ( pp(aa(fun(B,A),bool,order_mono(B,A),F3))
           => ( aa(A,B,G3,complete_lattice_gfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_adb(fun(A,B),fun(fun(B,A),fun(A,A)),G3),F3))) = complete_lattice_gfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_adc(fun(A,B),fun(fun(B,A),fun(B,B)),G3),F3)) ) ) ) ) ).

% gfp_rolling
tff(fact_7786_gfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] : complete_lattice_gfp(A,F3) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aqo(fun(A,A),fun(A,bool),F3))) ) ).

% gfp_def
tff(fact_7787_weak__coinduct__image,axiom,
    ! [A: $tType,B: $tType,A3: A,X6: set(A),G3: fun(A,B),F3: fun(set(B),set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),X6))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G3),X6)),aa(set(B),set(B),F3,aa(set(A),set(B),image2(A,B,G3),X6))))
       => pp(aa(set(B),bool,member(B,aa(A,B,G3,A3)),complete_lattice_gfp(set(B),F3))) ) ) ).

% weak_coinduct_image
tff(fact_7788_gfp__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,fun(A,A))] :
          ( ! [X3: A,Y3: A,W: A,Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),W)),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))) ) )
         => ( complete_lattice_gfp(A,aTP_Lamp_aqp(fun(A,fun(A,A)),fun(A,A),F3)) = complete_lattice_gfp(A,aTP_Lamp_ada(fun(A,fun(A,A)),fun(A,A),F3)) ) ) ) ).

% gfp_gfp
tff(fact_7789_weak__coinduct,axiom,
    ! [A: $tType,A3: A,X6: set(A),F3: fun(set(A),set(A))] :
      ( pp(aa(set(A),bool,member(A,A3),X6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,X6)))
       => pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3))) ) ) ).

% weak_coinduct
tff(fact_7790_gfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),G3: fun(A,A)] :
          ( ! [Z8: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,Z8)),aa(A,A,G3,Z8)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),complete_lattice_gfp(A,G3))) ) ) ).

% gfp_mono
tff(fact_7791_gfp__upperbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F3: fun(A,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,X6)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F3))) ) ) ).

% gfp_upperbound
tff(fact_7792_gfp__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),X6: A] :
          ( ! [U5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U5),aa(A,A,F3,U5)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U5),X6)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),X6)) ) ) ).

% gfp_least
tff(fact_7793_gfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),X: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F4))
         => ( ( aa(A,A,F4,X) = X )
           => ( ! [Z3: A] :
                  ( ( aa(A,A,F4,Z3) = Z3 )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X)) )
             => ( complete_lattice_gfp(A,F4) = X ) ) ) ) ) ).

% gfp_eqI
tff(fact_7794_def__coinduct__set,axiom,
    ! [A: $tType,A5: set(A),F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( ( A5 = complete_lattice_gfp(set(A),F3) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( pp(aa(set(A),bool,member(A,A3),X6))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),A5))))
           => pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ) ).

% def_coinduct_set
tff(fact_7795_coinduct__set,axiom,
    ! [A: $tType,F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
     => ( pp(aa(set(A),bool,member(A,A3),X6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),complete_lattice_gfp(set(A),F3)))))
         => pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3))) ) ) ) ).

% coinduct_set
tff(fact_7796_coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),X6: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3)))))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F3))) ) ) ) ).

% coinduct
tff(fact_7797_def__coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: A,F3: fun(A,A),X6: A] :
          ( ( A5 = complete_lattice_gfp(A,F3) )
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),A5))))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),A5)) ) ) ) ) ).

% def_coinduct
tff(fact_7798_coinduct__lemma,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F3: fun(A,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3)))))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3))),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3))))) ) ) ) ).

% coinduct_lemma
tff(fact_7799_gfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P: fun(A,bool)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ! [S8: A] :
                ( pp(aa(A,bool,P,S8))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),S8))
                 => pp(aa(A,bool,P,aa(A,A,F3,S8))) ) )
           => ( ! [M11: set(A)] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),M11))
                     => pp(aa(A,bool,P,X5)) )
                 => pp(aa(A,bool,P,aa(set(A),A,complete_Inf_Inf(A),M11))) )
             => pp(aa(A,bool,P,complete_lattice_gfp(A,F3))) ) ) ) ) ).

% gfp_ordinal_induct
tff(fact_7800_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N2)),F3)) = complete_lattice_gfp(A,F3) ) ) ) ).

% gfp_funpow
tff(fact_7801_lfp__le__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),complete_lattice_gfp(A,F3))) ) ) ).

% lfp_le_gfp
tff(fact_7802_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F3),top_top(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F3),top_top(A)) )
           => ( complete_lattice_gfp(A,F3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F3),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_7803_coinduct3,axiom,
    ! [A: $tType,F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
     => ( pp(aa(set(A),bool,member(A,A3),X6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_aqq(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),F3),X6)))))
         => pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3))) ) ) ) ).

% coinduct3
tff(fact_7804_AboveS__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] : order_AboveS(A,R3,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aqr(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A5)) ).

% AboveS_def
tff(fact_7805_wfP__SUP,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,fun(B,bool)))] :
      ( ! [I3: A] : wfP(B,aa(A,fun(B,fun(B,bool)),R3,I3))
     => ( ! [I3: A,J2: A] :
            ( ( aa(A,fun(B,fun(B,bool)),R3,I3) != aa(A,fun(B,fun(B,bool)),R3,J2) )
           => ( aa(fun(B,bool),fun(B,bool),aa(fun(B,bool),fun(fun(B,bool),fun(B,bool)),inf_inf(fun(B,bool)),aa(fun(B,fun(B,bool)),fun(B,bool),domainp(B,B),aa(A,fun(B,fun(B,bool)),R3,I3))),rangep(B,B,aa(A,fun(B,fun(B,bool)),R3,J2))) = bot_bot(fun(B,bool)) ) )
       => wfP(B,aa(set(fun(B,fun(B,bool))),fun(B,fun(B,bool)),complete_Sup_Sup(fun(B,fun(B,bool))),aa(set(A),set(fun(B,fun(B,bool))),image2(A,fun(B,fun(B,bool)),R3),top_top(set(A))))) ) ) ).

% wfP_SUP
tff(fact_7806_wfP__empty,axiom,
    ! [A: $tType] : wfP(A,aTP_Lamp_aqs(A,fun(A,bool))) ).

% wfP_empty
tff(fact_7807_wfP__subset,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),P3: fun(A,fun(A,bool))] :
      ( wfP(A,R3)
     => ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),P3),R3))
       => wfP(A,P3) ) ) ).

% wfP_subset
tff(fact_7808_Rangep_Ocases,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: B] :
      ( pp(aa(B,bool,rangep(A,B,R3),A3))
     => ~ ! [A6: A] : ~ pp(aa(B,bool,aa(A,fun(B,bool),R3,A6),A3)) ) ).

% Rangep.cases
tff(fact_7809_Rangep_Osimps,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: B] :
      ( pp(aa(B,bool,rangep(A,B,R3),A3))
    <=> ? [A8: A,B13: B] :
          ( ( A3 = B13 )
          & pp(aa(B,bool,aa(A,fun(B,bool),R3,A8),B13)) ) ) ).

% Rangep.simps
tff(fact_7810_RangePI,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B2))
     => pp(aa(B,bool,rangep(A,B,R3),B2)) ) ).

% RangePI
tff(fact_7811_RangepE,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),B2: B] :
      ( pp(aa(B,bool,rangep(A,B,R3),B2))
     => ~ ! [A6: A] : ~ pp(aa(B,bool,aa(A,fun(B,bool),R3,A6),B2)) ) ).

% RangepE
tff(fact_7812_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_7813_wfP__less,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wfP(A,ord_less(A)) ) ).

% wfP_less
tff(fact_7814_wfP__wf__eq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wfP(A,aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))
    <=> wf(A,R3) ) ).

% wfP_wf_eq
tff(fact_7815_wfP__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] :
      ( wfP(A,R3)
    <=> wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))) ) ).

% wfP_def
tff(fact_7816_AboveS__disjoint,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),order_AboveS(A,R3,A5)) = bot_bot(set(A)) ).

% AboveS_disjoint
tff(fact_7817_AboveS__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_AboveS(A,R3,A5)),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ).

% AboveS_Field
tff(fact_7818_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( order_AboveS(A,R3,B4) != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,member(A,B2),B4))
           => ( ( bNF_Wellorder_wo_suc(A,R3,B4) != B2 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R3,B4))),R3)) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_7819_wo__rel_Osuc__AboveS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( order_AboveS(A,R3,B4) != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_Wellorder_wo_suc(A,R3,B4)),order_AboveS(A,R3,B4))) ) ) ) ).

% wo_rel.suc_AboveS
tff(fact_7820_wo__rel_Osuc__least__AboveS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R3,B4)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R3,B4)),A3)),R3)) ) ) ).

% wo_rel.suc_least_AboveS
tff(fact_7821_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R3,B4)))
         => ( ! [A10: A] :
                ( pp(aa(set(A),bool,member(A,A10),order_AboveS(A,R3,B4)))
               => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A10)),R3)) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R3,B4) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_7822_wo__rel_Osuc__inField,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( order_AboveS(A,R3,B4) != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_Wellorder_wo_suc(A,R3,B4)),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ) ) ).

% wo_rel.suc_inField
tff(fact_7823_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A5)
       => ( ( order_AboveS(A,R3,A5) != bot_bot(set(A)) )
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R3,A5))),R3))
           => ( ( B2 != bNF_Wellorder_wo_suc(A,R3,A5) )
             => pp(aa(set(A),bool,member(A,B2),A5)) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_7824_right__total__relcompp__transfer,axiom,
    ! [A: $tType,E: $tType,C: $tType,F: $tType,B: $tType,D: $tType,B4: fun(A,fun(B,bool)),A5: fun(C,fun(D,bool)),C4: fun(E,fun(F,bool))] :
      ( right_total(A,B,B4)
     => pp(aa(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F,bool)),fun(D,fun(F,bool)))),bool,aa(fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),fun(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F,bool)),fun(D,fun(F,bool)))),bool),bNF_rel_fun(fun(C,fun(A,bool)),fun(D,fun(B,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool))),fun(fun(B,fun(F,bool)),fun(D,fun(F,bool))),bNF_rel_fun(C,D,fun(A,bool),fun(B,bool),A5,bNF_rel_fun(A,B,bool,bool,B4,fequal(bool))),bNF_rel_fun(fun(A,fun(E,bool)),fun(B,fun(F,bool)),fun(C,fun(E,bool)),fun(D,fun(F,bool)),bNF_rel_fun(A,B,fun(E,bool),fun(F,bool),B4,bNF_rel_fun(E,F,bool,bool,C4,fequal(bool))),bNF_rel_fun(C,D,fun(E,bool),fun(F,bool),A5,bNF_rel_fun(E,F,bool,bool,C4,fequal(bool))))),aTP_Lamp_aqt(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),B4)),relcompp(D,B,F))) ) ).

% right_total_relcompp_transfer
tff(fact_7825_relcompp__distrib,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(C,bool)),S: fun(C,fun(B,bool)),T2: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),aa(fun(C,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(fun(C,fun(B,bool)),fun(C,fun(B,bool))),sup_sup(fun(C,fun(B,bool))),S),T2)) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),S)),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),T2)) ).

% relcompp_distrib
tff(fact_7826_relcompp__distrib2,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: fun(A,fun(C,bool)),T2: fun(A,fun(C,bool)),R2: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(fun(A,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(C,bool)),fun(fun(A,fun(C,bool)),fun(A,fun(C,bool))),sup_sup(fun(A,fun(C,bool))),S),T2)),R2) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),S),R2)),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),T2),R2)) ).

% relcompp_distrib2
tff(fact_7827_relcompp__bot1,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),bot_bot(fun(A,fun(C,bool)))),R2) = bot_bot(fun(A,fun(B,bool))) ).

% relcompp_bot1
tff(fact_7828_relcompp__bot2,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: fun(A,fun(C,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),bot_bot(fun(C,fun(B,bool)))) = bot_bot(fun(A,fun(B,bool))) ).

% relcompp_bot2
tff(fact_7829_pos__fun__distr,axiom,
    ! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F: $tType,R2: fun(A,fun(E,bool)),S: fun(B,fun(F,bool)),R8: fun(E,fun(C,bool)),S3: fun(F,fun(D,bool))] : pp(aa(fun(fun(A,B),fun(fun(C,D),bool)),bool,aa(fun(fun(A,B),fun(fun(C,D),bool)),fun(fun(fun(A,B),fun(fun(C,D),bool)),bool),ord_less_eq(fun(fun(A,B),fun(fun(C,D),bool))),aa(fun(fun(E,F),fun(fun(C,D),bool)),fun(fun(A,B),fun(fun(C,D),bool)),aa(fun(fun(A,B),fun(fun(E,F),bool)),fun(fun(fun(E,F),fun(fun(C,D),bool)),fun(fun(A,B),fun(fun(C,D),bool))),relcompp(fun(A,B),fun(E,F),fun(C,D)),bNF_rel_fun(A,E,B,F,R2,S)),bNF_rel_fun(E,C,F,D,R8,S3))),bNF_rel_fun(A,C,B,D,aa(fun(E,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(E,bool)),fun(fun(E,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,E,C),R2),R8),aa(fun(F,fun(D,bool)),fun(B,fun(D,bool)),aa(fun(B,fun(F,bool)),fun(fun(F,fun(D,bool)),fun(B,fun(D,bool))),relcompp(B,F,D),S),S3)))) ).

% pos_fun_distr
tff(fact_7830_leq__OOI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( ( R2 = fequal(A) )
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R2),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R2),R2))) ) ).

% leq_OOI
tff(fact_7831_relcompp__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,R6: fun(A,fun(B,bool)),R3: fun(A,fun(B,bool)),S4: fun(B,fun(C,bool)),S2: fun(B,fun(C,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R6),R3))
     => ( pp(aa(fun(B,fun(C,bool)),bool,aa(fun(B,fun(C,bool)),fun(fun(B,fun(C,bool)),bool),ord_less_eq(fun(B,fun(C,bool))),S4),S2))
       => pp(aa(fun(A,fun(C,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(A,fun(C,bool)),bool),ord_less_eq(fun(A,fun(C,bool))),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R6),S4)),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S2))) ) ) ).

% relcompp_mono
tff(fact_7832_wo__rel_Oofilter__linord,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A5)
       => ( order_ofilter(A,R3,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5)) ) ) ) ) ).

% wo_rel.ofilter_linord
tff(fact_7833_relcompp__SUP__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S2: fun(A,fun(C,bool)),R3: fun(D,fun(C,fun(B,bool))),I5: set(D)] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),S2),aa(set(fun(C,fun(B,bool))),fun(C,fun(B,bool)),complete_Sup_Sup(fun(C,fun(B,bool))),aa(set(D),set(fun(C,fun(B,bool))),image2(D,fun(C,fun(B,bool)),R3),I5))) = aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(D),set(fun(A,fun(B,bool))),image2(D,fun(A,fun(B,bool)),aa(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool))),aTP_Lamp_aqu(fun(A,fun(C,bool)),fun(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool)))),S2),R3)),I5)) ).

% relcompp_SUP_distrib
tff(fact_7834_relcompp__SUP__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R3: fun(D,fun(A,fun(C,bool))),I5: set(D),S2: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(set(fun(A,fun(C,bool))),fun(A,fun(C,bool)),complete_Sup_Sup(fun(A,fun(C,bool))),aa(set(D),set(fun(A,fun(C,bool))),image2(D,fun(A,fun(C,bool)),R3),I5))),S2) = aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(D),set(fun(A,fun(B,bool))),image2(D,fun(A,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool))),aTP_Lamp_aqv(fun(D,fun(A,fun(C,bool))),fun(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool)))),R3),S2)),I5)) ).

% relcompp_SUP_distrib2
tff(fact_7835_pcr__Domainp__par,axiom,
    ! [A: $tType,B: $tType,C: $tType,B4: fun(A,fun(B,bool)),P24: fun(A,bool),A5: fun(C,fun(A,bool)),P1: fun(C,bool),P23: fun(C,bool)] :
      ( ( aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),B4) = P24 )
     => ( ( aa(fun(C,fun(A,bool)),fun(C,bool),domainp(C,A),A5) = P1 )
       => ( pp(aa(fun(A,bool),bool,aa(fun(C,bool),fun(fun(A,bool),bool),bNF_rel_fun(C,A,bool,bool,A5,fequal(bool)),P23),P24))
         => ( aa(fun(C,fun(B,bool)),fun(C,bool),domainp(C,B),aa(fun(A,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,A,B),A5),B4)) = aa(fun(C,bool),fun(C,bool),aa(fun(C,bool),fun(fun(C,bool),fun(C,bool)),inf_inf(fun(C,bool)),P1),P23) ) ) ) ) ).

% pcr_Domainp_par
tff(fact_7836_nchotomy__relcomppE,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: fun(B,A),R3: fun(C,fun(A,bool)),S2: fun(A,fun(D,bool)),A3: C,C2: D] :
      ( ! [Y3: A] :
        ? [X5: B] : Y3 = aa(B,A,F3,X5)
     => ( pp(aa(D,bool,aa(C,fun(D,bool),aa(fun(A,fun(D,bool)),fun(C,fun(D,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(D,bool)),fun(C,fun(D,bool))),relcompp(C,A,D),R3),S2),A3),C2))
       => ~ ! [B5: B] :
              ( pp(aa(A,bool,aa(C,fun(A,bool),R3,A3),aa(B,A,F3,B5)))
             => ~ pp(aa(D,bool,aa(A,fun(D,bool),S2,aa(B,A,F3,B5)),C2)) ) ) ) ).

% nchotomy_relcomppE
tff(fact_7837_relcompp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool)),A1: A,A22: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S2),A1),A22))
     => ~ ! [B5: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),R3,A1),B5))
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),S2,B5),A22)) ) ) ).

% relcompp.cases
tff(fact_7838_relcompp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool)),A1: A,A22: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S2),A1),A22))
    <=> ? [A8: A,B13: B,C5: C] :
          ( ( A1 = A8 )
          & ( A22 = C5 )
          & pp(aa(B,bool,aa(A,fun(B,bool),R3,A8),B13))
          & pp(aa(C,bool,aa(B,fun(C,bool),S2,B13),C5)) ) ) ).

% relcompp.simps
tff(fact_7839_OO__eq,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : aa(fun(B,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,B,B),R2),fequal(B)) = R2 ).

% OO_eq
tff(fact_7840_eq__OO,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,A,B),fequal(A)),R2) = R2 ).

% eq_OO
tff(fact_7841_relcomppE,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool)),A3: A,C2: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S2),A3),C2))
     => ~ ! [B5: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B5))
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),S2,B5),C2)) ) ) ).

% relcomppE
tff(fact_7842_relcomppI,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(B,bool)),A3: A,B2: B,S2: fun(B,fun(C,bool)),C2: C] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B2))
     => ( pp(aa(C,bool,aa(B,fun(C,bool),S2,B2),C2))
       => pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S2),A3),C2)) ) ) ).

% relcomppI
tff(fact_7843_relcompp__apply,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(B,bool)),S: fun(B,fun(C,bool)),A3: A,C2: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R2),S),A3),C2))
    <=> ? [B13: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,A3),B13))
          & pp(aa(C,bool,aa(B,fun(C,bool),S,B13),C2)) ) ) ).

% relcompp_apply
tff(fact_7844_relcompp__assoc,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,R3: fun(A,fun(D,bool)),S2: fun(D,fun(C,bool)),T6: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(fun(D,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(D,bool)),fun(fun(D,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,D,C),R3),S2)),T6) = aa(fun(D,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(D,bool)),fun(fun(D,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,D,B),R3),aa(fun(C,fun(B,bool)),fun(D,fun(B,bool)),aa(fun(D,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(D,fun(B,bool))),relcompp(D,C,B),S2),T6)) ).

% relcompp_assoc
tff(fact_7845_OO__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: fun(A,fun(C,bool)),S: fun(C,fun(B,bool)),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),S),X5),Xa3))
    <=> ? [Y5: C] :
          ( pp(aa(C,bool,aa(A,fun(C,bool),R2,X5),Y5))
          & pp(aa(B,bool,aa(C,fun(B,bool),S,Y5),Xa3)) ) ) ).

% OO_def
tff(fact_7846_pcr__Domainp,axiom,
    ! [C: $tType,B: $tType,A: $tType,B4: fun(A,fun(B,bool)),P: fun(A,bool),A5: fun(C,fun(A,bool))] :
      ( ( aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),B4) = P )
     => ! [X5: C] :
          ( pp(aa(C,bool,aa(fun(C,fun(B,bool)),fun(C,bool),domainp(C,B),aa(fun(A,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,A,B),A5),B4)),X5))
        <=> ? [Y5: A] :
              ( pp(aa(A,bool,aa(C,fun(A,bool),A5,X5),Y5))
              & pp(aa(A,bool,P,Y5)) ) ) ) ).

% pcr_Domainp
tff(fact_7847_relcomp__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,X5: set(product_prod(A,B)),Xa3: set(product_prod(B,C))] : relcomp(A,B,C,X5,Xa3) = aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)),aTP_Lamp_aqw(set(product_prod(B,C)),fun(B,fun(C,bool)),Xa3)))) ).

% relcomp_def
tff(fact_7848_relcompp__relcomp__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(B,C)),X5: A,Xa3: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),aTP_Lamp_aqw(set(product_prod(B,C)),fun(B,fun(C,bool)),S2)),X5),Xa3))
    <=> pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X5),Xa3)),relcomp(A,B,C,R3,S2))) ) ).

% relcompp_relcomp_eq
tff(fact_7849_wo__rel_Oofilter__UNION,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(A))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),I5))
           => order_ofilter(A,R3,aa(B,set(A),A5,I3)) )
       => order_ofilter(A,R3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))) ) ) ).

% wo_rel.ofilter_UNION
tff(fact_7850_wo__rel_Oofilter__AboveS__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A5)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),order_AboveS(A,R3,A5)) = aa(set(product_prod(A,A)),set(A),field2(A),R3) ) ) ) ).

% wo_rel.ofilter_AboveS_Field
tff(fact_7851_ofilterIncl__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R3) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_aqx(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R3))) ).

% ofilterIncl_def
tff(fact_7852_neg__fun__distr1,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,E: $tType,F: $tType,R2: fun(A,fun(B,bool)),R8: fun(B,fun(C,bool)),S: fun(D,fun(F,bool)),S3: fun(F,fun(E,bool))] :
      ( left_unique(A,B,R2)
     => ( right_total(A,B,R2)
       => ( right_unique(B,C,R8)
         => ( left_total(B,C,R8)
           => pp(aa(fun(fun(A,D),fun(fun(C,E),bool)),bool,aa(fun(fun(A,D),fun(fun(C,E),bool)),fun(fun(fun(A,D),fun(fun(C,E),bool)),bool),ord_less_eq(fun(fun(A,D),fun(fun(C,E),bool))),bNF_rel_fun(A,C,D,E,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R2),R8),aa(fun(F,fun(E,bool)),fun(D,fun(E,bool)),aa(fun(D,fun(F,bool)),fun(fun(F,fun(E,bool)),fun(D,fun(E,bool))),relcompp(D,F,E),S),S3))),aa(fun(fun(B,F),fun(fun(C,E),bool)),fun(fun(A,D),fun(fun(C,E),bool)),aa(fun(fun(A,D),fun(fun(B,F),bool)),fun(fun(fun(B,F),fun(fun(C,E),bool)),fun(fun(A,D),fun(fun(C,E),bool))),relcompp(fun(A,D),fun(B,F),fun(C,E)),bNF_rel_fun(A,B,D,F,R2,S)),bNF_rel_fun(B,C,F,E,R8,S3)))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_7853_typedef__right__unique,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T2: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T2,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => right_unique(A,B,T2) ) ) ).

% typedef_right_unique
tff(fact_7854_neg__fun__distr2,axiom,
    ! [F: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R8: fun(A,fun(B,bool)),S3: fun(C,fun(D,bool)),R2: fun(E,fun(A,bool)),S: fun(F,fun(C,bool))] :
      ( right_unique(A,B,R8)
     => ( left_total(A,B,R8)
       => ( left_unique(C,D,S3)
         => ( right_total(C,D,S3)
           => pp(aa(fun(fun(E,F),fun(fun(B,D),bool)),bool,aa(fun(fun(E,F),fun(fun(B,D),bool)),fun(fun(fun(E,F),fun(fun(B,D),bool)),bool),ord_less_eq(fun(fun(E,F),fun(fun(B,D),bool))),bNF_rel_fun(E,B,F,D,aa(fun(A,fun(B,bool)),fun(E,fun(B,bool)),aa(fun(E,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(E,fun(B,bool))),relcompp(E,A,B),R2),R8),aa(fun(C,fun(D,bool)),fun(F,fun(D,bool)),aa(fun(F,fun(C,bool)),fun(fun(C,fun(D,bool)),fun(F,fun(D,bool))),relcompp(F,C,D),S),S3))),aa(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(E,F),fun(fun(B,D),bool)),aa(fun(fun(E,F),fun(fun(A,C),bool)),fun(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(E,F),fun(fun(B,D),bool))),relcompp(fun(E,F),fun(A,C),fun(B,D)),bNF_rel_fun(E,A,F,C,R2,S)),bNF_rel_fun(A,B,C,D,R8,S3)))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_7855_composed__equiv__rel__eq__onp,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),P: fun(A,bool),P6: fun(B,bool),P9: fun(A,bool)] :
      ( left_unique(A,B,R2)
     => ( pp(aa(fun(B,bool),bool,aa(fun(A,bool),fun(fun(B,bool),bool),bNF_rel_fun(A,B,bool,bool,R2,fequal(bool)),P),P6))
       => ( ( aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R2) = P9 )
         => ( aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R2),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),bNF_eq_onp(B,P6)),conversep(A,B,R2))) = bNF_eq_onp(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),P9),P)) ) ) ) ) ).

% composed_equiv_rel_eq_onp
tff(fact_7856_wo__rel_Oofilter__under__UNION,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A5)
       => ( A5 = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),order_under(A,R3)),A5)) ) ) ) ).

% wo_rel.ofilter_under_UNION
tff(fact_7857_conversep__eq,axiom,
    ! [A: $tType] : conversep(A,A,fequal(A)) = fequal(A) ).

% conversep_eq
tff(fact_7858_conversep__iff,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: B,B2: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R3),A3),B2))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),R3,B2),A3)) ) ).

% conversep_iff
tff(fact_7859_conversep__inject,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,fun(A,bool)),S2: fun(B,fun(A,bool))] :
      ( ( conversep(B,A,R3) = conversep(B,A,S2) )
    <=> ( R3 = S2 ) ) ).

% conversep_inject
tff(fact_7860_conversep__conversep,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : conversep(B,A,conversep(A,B,R3)) = R3 ).

% conversep_conversep
tff(fact_7861_conversep__noteq,axiom,
    ! [A: $tType,X5: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),conversep(A,A,aTP_Lamp_sn(A,fun(A,bool))),X5),Xa3))
    <=> ( X5 != Xa3 ) ) ).

% conversep_noteq
tff(fact_7862_conversep__mono,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,fun(A,bool)),S2: fun(B,fun(A,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),conversep(B,A,R3)),conversep(B,A,S2)))
    <=> pp(aa(fun(B,fun(A,bool)),bool,aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),bool),ord_less_eq(fun(B,fun(A,bool))),R3),S2)) ) ).

% conversep_mono
tff(fact_7863_left__unique__alt__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( left_unique(A,B,R2)
    <=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R2),conversep(A,B,R2))),fequal(A))) ) ).

% left_unique_alt_def
tff(fact_7864_Quotient__composition__le__eq,axiom,
    ! [B: $tType,A: $tType,T2: fun(A,fun(B,bool)),R2: fun(B,fun(B,bool))] :
      ( left_unique(A,B,T2)
     => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),R2),fequal(B)))
       => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T2),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),R2),conversep(A,B,T2)))),fequal(A))) ) ) ).

% Quotient_composition_le_eq
tff(fact_7865_left__total__alt__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( left_total(A,B,R2)
    <=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R2),conversep(A,B,R2)))) ) ).

% left_total_alt_def
tff(fact_7866_Quotient__composition__ge__eq,axiom,
    ! [B: $tType,A: $tType,T2: fun(A,fun(B,bool)),R2: fun(B,fun(B,bool))] :
      ( left_total(A,B,T2)
     => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),fequal(B)),R2))
       => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T2),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),R2),conversep(A,B,T2))))) ) ) ).

% Quotient_composition_ge_eq
tff(fact_7867_right__unique__alt__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] :
      ( right_unique(A,B,R2)
    <=> pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),aa(fun(A,fun(B,bool)),fun(B,fun(B,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(B,fun(B,bool))),relcompp(B,A,B),conversep(A,B,R2)),R2)),fequal(B))) ) ).

% right_unique_alt_def
tff(fact_7868_right__total__alt__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( right_total(A,B,R2)
    <=> pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),fequal(B)),aa(fun(A,fun(B,bool)),fun(B,fun(B,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(B,fun(B,bool))),relcompp(B,A,B),conversep(A,B,R2)),R2))) ) ).

% right_total_alt_def
tff(fact_7869_converse__relcompp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: fun(B,fun(C,bool)),S2: fun(C,fun(A,bool))] : conversep(B,A,aa(fun(C,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(C,bool)),fun(fun(C,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,C,A),R3),S2)) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),conversep(C,A,S2)),conversep(B,C,R3)) ).

% converse_relcompp
tff(fact_7870_leq__conversepI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( ( R2 = fequal(A) )
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R2),conversep(A,A,R2))) ) ).

% leq_conversepI
tff(fact_7871_conversep__le__swap,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),S2: fun(B,fun(A,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R3),conversep(B,A,S2)))
    <=> pp(aa(fun(B,fun(A,bool)),bool,aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),bool),ord_less_eq(fun(B,fun(A,bool))),conversep(A,B,R3)),S2)) ) ).

% conversep_le_swap
tff(fact_7872_under__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),A3)),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ).

% under_Field
tff(fact_7873_converse__meet,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,fun(A,bool)),S2: fun(B,fun(A,bool))] : conversep(B,A,aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),inf_inf(fun(B,fun(A,bool))),R3),S2)) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),conversep(B,A,R3)),conversep(B,A,S2)) ).

% converse_meet
tff(fact_7874_conversep_Osimps,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A1: B,A22: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R3),A1),A22))
    <=> ? [A8: A,B13: B] :
          ( ( A1 = B13 )
          & ( A22 = A8 )
          & pp(aa(B,bool,aa(A,fun(B,bool),R3,A8),B13)) ) ) ).

% conversep.simps
tff(fact_7875_conversepD,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),B2: B,A3: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R3),B2),A3))
     => pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B2)) ) ).

% conversepD
tff(fact_7876_conversepE,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A1: B,A22: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R3),A1),A22))
     => pp(aa(B,bool,aa(A,fun(B,bool),R3,A22),A1)) ) ).

% conversepE
tff(fact_7877_conversepI,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B2))
     => pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R3),B2),A3)) ) ).

% conversepI
tff(fact_7878_converse__join,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,fun(A,bool)),S2: fun(B,fun(A,bool))] : conversep(B,A,aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),sup_sup(fun(B,fun(A,bool))),R3),S2)) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),conversep(B,A,R3)),conversep(B,A,S2)) ).

% converse_join
tff(fact_7879_conversep__converse__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),X5: B,Xa3: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),X5),Xa3))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X5),Xa3)),converse(A,B,R3))) ) ).

% conversep_converse_eq
tff(fact_7880_under__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : aa(A,set(A),order_under(A,R3),A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_alg(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),A3)) ).

% under_def
tff(fact_7881_converse__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : converse(A,B,X5) = aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)))) ).

% converse_def
tff(fact_7882_under__incr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),A3)),aa(A,set(A),order_under(A,R3),B2))) ) ) ).

% under_incr
tff(fact_7883_ofilter__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_ofilter(A,R3,A5)
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),X4)),A5)) ) ) ) ).

% ofilter_def
tff(fact_7884_flip__pred,axiom,
    ! [A: $tType,B: $tType,A5: set(product_prod(A,B)),R2: fun(B,fun(A,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),conversep(B,A,R2)))))
     => pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),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_wx(A,fun(B,product_prod(B,A))))),A5)),aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),R2)))) ) ).

% flip_pred
tff(fact_7885_wo__rel_Oofilter__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A5)
      <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
          & ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),X4)),A5)) ) ) ) ) ).

% wo_rel.ofilter_def
tff(fact_7886_multiset_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : rel_mset(A,B,R2) = aa(fun(multiset(product_prod(A,B)),fun(multiset(B),bool)),fun(multiset(A),fun(multiset(B),bool)),aa(fun(multiset(A),fun(multiset(product_prod(A,B)),bool)),fun(fun(multiset(product_prod(A,B)),fun(multiset(B),bool)),fun(multiset(A),fun(multiset(B),bool))),relcompp(multiset(A),multiset(product_prod(A,B)),multiset(B)),conversep(multiset(product_prod(A,B)),multiset(A),bNF_Grp(multiset(product_prod(A,B)),multiset(A),aa(fun(multiset(product_prod(A,B)),bool),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_alj(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),R2)),image_mset(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(multiset(product_prod(A,B)),multiset(B),aa(fun(multiset(product_prod(A,B)),bool),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_alj(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),R2)),image_mset(product_prod(A,B),B,product_snd(A,B)))) ).

% multiset.rel_compp_Grp
tff(fact_7887_fun_Orel__compp__Grp,axiom,
    ! [D: $tType,B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : bNF_rel_fun(D,D,A,B,fequal(D),R2) = aa(fun(fun(D,product_prod(A,B)),fun(fun(D,B),bool)),fun(fun(D,A),fun(fun(D,B),bool)),aa(fun(fun(D,A),fun(fun(D,product_prod(A,B)),bool)),fun(fun(fun(D,product_prod(A,B)),fun(fun(D,B),bool)),fun(fun(D,A),fun(fun(D,B),bool))),relcompp(fun(D,A),fun(D,product_prod(A,B)),fun(D,B)),conversep(fun(D,product_prod(A,B)),fun(D,A),bNF_Grp(fun(D,product_prod(A,B)),fun(D,A),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_ahf(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R2)),comp(product_prod(A,B),A,D,product_fst(A,B))))),bNF_Grp(fun(D,product_prod(A,B)),fun(D,B),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_ahf(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R2)),comp(product_prod(A,B),B,D,product_snd(A,B)))) ).

% fun.rel_compp_Grp
tff(fact_7888_fun_Orel__Grp,axiom,
    ! [D: $tType,B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] : bNF_rel_fun(D,D,A,B,fequal(D),bNF_Grp(A,B,A5,F3)) = bNF_Grp(fun(D,A),fun(D,B),aa(fun(fun(D,A),bool),set(fun(D,A)),collect(fun(D,A)),aTP_Lamp_aqy(set(A),fun(fun(D,A),bool),A5)),comp(A,B,D,F3)) ).

% fun.rel_Grp
tff(fact_7889_list_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] : list_all2(A,B,bNF_Grp(A,B,A5,F3)) = bNF_Grp(list(A),list(B),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_ajf(set(A),fun(list(A),bool),A5)),map(A,B,F3)) ).

% list.rel_Grp
tff(fact_7890_Grp__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bNF_Grp(A,B,A5,F3),X5),Xa3))
    <=> ( ( Xa3 = aa(A,B,F3,X5) )
        & pp(aa(set(A),bool,member(A,X5),A5)) ) ) ).

% Grp_def
tff(fact_7891_Grp__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),bNF_Grp(A,B,A5,F3)),bNF_Grp(A,B,B4,F3))) ) ).

% Grp_mono
tff(fact_7892_multiset_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] : rel_mset(A,B,bNF_Grp(A,B,A5,F3)) = bNF_Grp(multiset(A),multiset(B),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aTP_Lamp_aqz(set(A),fun(multiset(A),bool),A5)),image_mset(A,B,F3)) ).

% multiset.rel_Grp
tff(fact_7893_eq__le__Grp__id__iff,axiom,
    ! [A: $tType,R2: fun(A,bool)] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),bNF_Grp(A,A,aa(fun(A,bool),set(A),collect(A),R2),id(A))))
    <=> ! [X_12: A] : pp(aa(A,bool,R2,X_12)) ) ).

% eq_le_Grp_id_iff
tff(fact_7894_OO__Grp__alt,axiom,
    ! [A: $tType,C: $tType,B: $tType,A5: set(C),F3: fun(C,A),G3: fun(C,B),X5: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),conversep(C,A,bNF_Grp(C,A,A5,F3))),bNF_Grp(C,B,A5,G3)),X5),Xa3))
    <=> ? [Z6: C] :
          ( pp(aa(set(C),bool,member(C,Z6),A5))
          & ( aa(C,A,F3,Z6) = X5 )
          & ( aa(C,B,G3,Z6) = Xa3 ) ) ) ).

% OO_Grp_alt
tff(fact_7895_list_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : list_all2(A,B,R2) = aa(fun(list(product_prod(A,B)),fun(list(B),bool)),fun(list(A),fun(list(B),bool)),aa(fun(list(A),fun(list(product_prod(A,B)),bool)),fun(fun(list(product_prod(A,B)),fun(list(B),bool)),fun(list(A),fun(list(B),bool))),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)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ako(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R2)),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)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ako(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R2)),map(product_prod(A,B),B,product_snd(A,B)))) ).

% list.rel_compp_Grp
tff(fact_7896_bsqr__ofilter,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),D4: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R3),D4)
       => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less(set(product_prod(A,A))),D4),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_aje(set(product_prod(A,A)),fun(A,set(A)),R3))))
         => ( ~ ? [A6: A] : aa(set(product_prod(A,A)),set(A),field2(A),R3) = aa(A,set(A),order_under(A,R3),A6)
           => ? [A11: set(A)] :
                ( order_ofilter(A,R3,A11)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A11),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
                & pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),D4),product_Sigma(A,A,A11,aTP_Lamp_yj(set(A),fun(A,set(A)),A11)))) ) ) ) ) ) ).

% bsqr_ofilter
tff(fact_7897_Well__order__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => order_well_order_on(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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ).

% Well_order_Restr
tff(fact_7898_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_7899_well__order__on__domain,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_well_order_on(A,A5,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( pp(aa(set(A),bool,member(A,A3),A5))
          & pp(aa(set(A),bool,member(A,B2),A5)) ) ) ) ).

% well_order_on_domain
tff(fact_7900_natLeq__on__well__order__on,axiom,
    ! [N2: nat] : order_well_order_on(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ajh(nat,fun(nat,fun(nat,bool)),N2)))) ).

% natLeq_on_well_order_on
tff(fact_7901_well__order__on__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => order_well_order_on(A,A5,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) ) ) ).

% well_order_on_Restr
tff(fact_7902_Field__Restr__ofilter,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( 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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))) = A5 ) ) ) ).

% Field_Restr_ofilter
tff(fact_7903_natLeq__on__Well__order,axiom,
    ! [N2: nat] : order_well_order_on(nat,aa(set(product_prod(nat,nat)),set(nat),field2(nat),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ajh(nat,fun(nat,fun(nat,bool)),N2)))),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ajh(nat,fun(nat,fun(nat,bool)),N2)))) ).

% natLeq_on_Well_order
tff(fact_7904_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
      <=> ! [A13: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A13),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
           => ( ( A13 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A13))
                  & ! [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),A13))
                     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R3)) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_7905_ofilter__Restr__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
         => order_ofilter(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))),A5) ) ) ) ).

% ofilter_Restr_subset
tff(fact_7906_ofilter__Restr__Int,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => order_ofilter(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ) ) ).

% ofilter_Restr_Int
tff(fact_7907_bsqr__max2,axiom,
    ! [A: $tType,R3: 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),R3),R3)
     => ( pp(aa(set(product_prod(product_prod(A,A),product_prod(A,A))),bool,member(product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R3,A1,A22)),bNF_We1388413361240627857o_max2(A,R3,B1,B22))),R3)) ) ) ).

% bsqr_max2
tff(fact_7908_ofilter__Restr__under,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),A3: A] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( aa(A,set(A),order_under(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),A3) = aa(A,set(A),order_under(A,R3),A3) ) ) ) ) ).

% ofilter_Restr_under
tff(fact_7909_UNION__inj__on__ofilter,axiom,
    ! [C: $tType,A: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(A)),F3: fun(A,C)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),I5))
           => order_ofilter(A,R3,aa(B,set(A),A5,I3)) )
       => ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),I5))
             => inj_on(A,C,F3,aa(B,set(A),A5,I3)) )
         => inj_on(A,C,F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))) ) ) ) ).

% UNION_inj_on_ofilter
tff(fact_7910_ofilter__subset__embedS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( order_ofilter(A,R3,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
          <=> 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))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))),id(A)) ) ) ) ) ).

% ofilter_subset_embedS
tff(fact_7911_embedS__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Wellorder_embedS(A,B,R3,R6,F3)
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(set(product_prod(A,A)),set(A),field2(A),R3))),aa(set(product_prod(B,B)),set(B),field2(B),R6))) ) ) ).

% embedS_Field
tff(fact_7912_ofilter__subset__embedS__iso,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( order_ofilter(A,R3,B4)
         => ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
            <=> 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))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))),id(A)) )
            & ( ( A5 = B4 )
            <=> 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))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))),id(A)) ) ) ) ) ) ).

% ofilter_subset_embedS_iso
tff(fact_7913_ofilter__subset__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( order_ofilter(A,R3,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B4))
          <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ) ).

% ofilter_subset_ordLess
tff(fact_7914_iso__forward,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
     => ( bNF_Wellorder_iso(A,B,R3,R6,F3)
       => pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,X)),aa(A,B,F3,Y))),R6)) ) ) ).

% iso_forward
tff(fact_7915_ordLess__irreflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R3)),bNF_We4044943003108391690rdLess(A,A))) ).

% ordLess_irreflexive
tff(fact_7916_ordLess__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_We4044943003108391690rdLess(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLess_transitive
tff(fact_7917_iso__iff2,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( bNF_Wellorder_iso(A,B,R3,R6,F3)
    <=> ( bij_betw(A,B,F3,aa(set(product_prod(A,A)),set(A),field2(A),R3),aa(set(product_prod(B,B)),set(B),field2(B),R6))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
           => ! [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R3))
                <=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,X4)),aa(A,B,F3,Xa))),R6)) ) ) ) ) ) ).

% iso_iff2
tff(fact_7918_finite__ordLess__infinite,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(set(product_prod(B,B)),set(B),field2(B),R6)))
           => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).

% finite_ordLess_infinite
tff(fact_7919_ordLess__def,axiom,
    ! [A2: $tType,A: $tType] : bNF_We4044943003108391690rdLess(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_ara(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordLess_def
tff(fact_7920_ofilter__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),R3)),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).

% ofilter_ordLess
tff(fact_7921_underS__Restr__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R3) != bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,A3),aa(A,fun(A,set(A)),aTP_Lamp_arb(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),A3)))),R3)),bNF_We4044943003108391690rdLess(A,A))) ) ) ).

% underS_Restr_ordLess
tff(fact_7922_ofilter__embed,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
      <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
          & bNF_Wellorder_embed(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))),R3,id(A)) ) ) ) ).

% ofilter_embed
tff(fact_7923_ordLess__not__embed,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => ~ ? [X_13: fun(B,A)] : bNF_Wellorder_embed(B,A,R6,R3,X_13) ) ).

% ordLess_not_embed
tff(fact_7924_underS__Field2,axiom,
    ! [A: $tType,A3: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R3,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ).

% underS_Field2
tff(fact_7925_underS__subset__under,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),aa(A,set(A),order_under(A,R3),A3))) ).

% underS_subset_under
tff(fact_7926_underS__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : order_underS(A,R3,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_arc(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),A3)) ).

% underS_def
tff(fact_7927_underS__E,axiom,
    ! [A: $tType,I2: A,R2: set(product_prod(A,A)),J: A] :
      ( pp(aa(set(A),bool,member(A,I2),order_underS(A,R2,J)))
     => ( ( I2 != J )
        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J)),R2)) ) ) ).

% underS_E
tff(fact_7928_underS__I,axiom,
    ! [A: $tType,I2: A,J: A,R2: set(product_prod(A,A))] :
      ( ( I2 != J )
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J)),R2))
       => pp(aa(set(A),bool,member(A,I2),order_underS(A,R2,J))) ) ) ).

% underS_I
tff(fact_7929_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [A: $tType,I2: A,R2: set(product_prod(A,A)),J: A] :
      ( pp(aa(set(A),bool,member(A,I2),order_underS(A,R2,J)))
     => pp(aa(set(A),bool,member(A,I2),aa(set(product_prod(A,A)),set(A),field2(A),R2))) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_7930_Order__Relation_OunderS__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ).

% Order_Relation.underS_Field
tff(fact_7931_underS__empty,axiom,
    ! [A: $tType,A3: A,R3: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( order_underS(A,R3,A3) = bot_bot(set(A)) ) ) ).

% underS_empty
tff(fact_7932_embed__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( bNF_Wellorder_embed(A,B,R3,R6,F3)
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(set(product_prod(A,A)),set(A),field2(A),R3))),aa(set(product_prod(B,B)),set(B),field2(B),R6))) ) ).

% embed_Field
tff(fact_7933_underS__Field3,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( ( aa(set(product_prod(A,A)),set(A),field2(A),R3) != bot_bot(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R3,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ).

% underS_Field3
tff(fact_7934_iso__defs_I2_J,axiom,
    ! [A: $tType,A2: $tType,X5: set(product_prod(A,A)),Xa3: set(product_prod(A2,A2)),Xb2: fun(A,A2)] :
      ( bNF_Wellorder_iso(A,A2,X5,Xa3,Xb2)
    <=> ( bNF_Wellorder_embed(A,A2,X5,Xa3,Xb2)
        & bij_betw(A,A2,Xb2,aa(set(product_prod(A,A)),set(A),field2(A),X5),aa(set(product_prod(A2,A2)),set(A2),field2(A2),Xa3)) ) ) ).

% iso_defs(2)
tff(fact_7935_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R12),R23)),bNF_We4044943003108391690rdLess(A,B)))
     => ( bNF_Wellorder_embed(A,B,R12,R23,F3)
       => ( aa(set(A),set(B),image2(A,B,F3),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_7936_embedS__defs_I2_J,axiom,
    ! [A: $tType,A2: $tType,X5: set(product_prod(A,A)),Xa3: set(product_prod(A2,A2)),Xb2: fun(A,A2)] :
      ( bNF_Wellorder_embedS(A,A2,X5,Xa3,Xb2)
    <=> ( bNF_Wellorder_embed(A,A2,X5,Xa3,Xb2)
        & ~ bij_betw(A,A2,Xb2,aa(set(product_prod(A,A)),set(A),field2(A),X5),aa(set(product_prod(A2,A2)),set(A2),field2(A2),Xa3)) ) ) ).

% embedS_defs(2)
tff(fact_7937_ordLess__iff,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
    <=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
        & order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
        & ~ ? [X_12: fun(B,A)] : bNF_Wellorder_embed(B,A,R6,R3,X_12) ) ) ).

% ordLess_iff
tff(fact_7938_underS__incl__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),order_underS(A,R3,B2)))
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ) ).

% underS_incl_iff
tff(fact_7939_underS__incr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R3)
     => ( antisym(A,R3)
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),order_underS(A,R3,B2))) ) ) ) ).

% underS_incr
tff(fact_7940_embed__defs_I2_J,axiom,
    ! [A2: $tType,A: $tType,X5: set(product_prod(A,A)),Xa3: set(product_prod(A2,A2)),Xb2: fun(A,A2)] :
      ( bNF_Wellorder_embed(A,A2,X5,Xa3,Xb2)
    <=> ! [Xc2: A] :
          ( pp(aa(set(A),bool,member(A,Xc2),aa(set(product_prod(A,A)),set(A),field2(A),X5)))
         => bij_betw(A,A2,Xb2,aa(A,set(A),order_under(A,X5),Xc2),aa(A2,set(A2),order_under(A2,Xa3),aa(A,A2,Xb2,Xc2))) ) ) ).

% embed_defs(2)
tff(fact_7941_embedS__iff,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Wellorder_embed(A,B,R3,R6,F3)
       => ( bNF_Wellorder_embedS(A,B,R3,R6,F3)
        <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(set(product_prod(A,A)),set(A),field2(A),R3))),aa(set(product_prod(B,B)),set(B),field2(B),R6))) ) ) ) ).

% embedS_iff
tff(fact_7942_embed__implies__iso__Restr,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(B,A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( bNF_Wellorder_embed(B,A,R6,R3,F3)
         => bNF_Wellorder_iso(B,A,R6,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F3),aa(set(product_prod(B,B)),set(B),field2(B),R6)),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_ard(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),R6),F3))),F3) ) ) ) ).

% embed_implies_iso_Restr
tff(fact_7943_embed__ordLess__ofilterIncl,axiom,
    ! [B: $tType,A: $tType,C: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(C,C)),F132: fun(A,C),F232: fun(B,C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R12),R23)),bNF_We4044943003108391690rdLess(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R23),R32)),bNF_We4044943003108391690rdLess(B,C)))
       => ( bNF_Wellorder_embed(A,C,R12,R32,F132)
         => ( bNF_Wellorder_embed(B,C,R23,R32,F232)
           => pp(aa(set(product_prod(set(C),set(C))),bool,member(product_prod(set(C),set(C)),aa(set(C),product_prod(set(C),set(C)),aa(set(C),fun(set(C),product_prod(set(C),set(C))),product_Pair(set(C),set(C)),aa(set(A),set(C),image2(A,C,F132),aa(set(product_prod(A,A)),set(A),field2(A),R12))),aa(set(B),set(C),image2(B,C,F232),aa(set(product_prod(B,B)),set(B),field2(B),R23)))),bNF_We413866401316099525erIncl(C,R32))) ) ) ) ) ).

% embed_ordLess_ofilterIncl
tff(fact_7944_Refl__under__underS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( aa(A,set(A),order_under(A,R3),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R3,A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_7945_ofilter__subset__embed,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( order_ofilter(A,R3,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
          <=> bNF_Wellorder_embed(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))),id(A)) ) ) ) ) ).

% ofilter_subset_embed
tff(fact_7946_ordLess__iff__ordIso__Restr,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_We4044943003108391690rdLess(B,A)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
              & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_arb(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4))))),bNF_Wellorder_ordIso(B,A))) ) ) ) ) ).

% ordLess_iff_ordIso_Restr
tff(fact_7947_ordLeq__iff__ordLess__Restr,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
             => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_arb(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4)))),R6)),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).

% ordLeq_iff_ordLess_Restr
tff(fact_7948_exists__minim__Well__order,axiom,
    ! [A: $tType,R2: set(set(product_prod(A,A)))] :
      ( ( R2 != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X3),R2))
           => order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),X3),X3) )
       => ? [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X3),R2))
            & ! [Xa3: set(product_prod(A,A))] :
                ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),Xa3),R2))
               => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% exists_minim_Well_order
tff(fact_7949_Well__order__iso__copy,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(A,A)),F3: fun(A,B),A15: set(B)] :
      ( order_well_order_on(A,A5,R3)
     => ( bij_betw(A,B,F3,A5,A15)
       => ? [R13: set(product_prod(B,B))] :
            ( order_well_order_on(B,A15,R13)
            & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R13)),bNF_Wellorder_ordIso(A,B))) ) ) ) ).

% Well_order_iso_copy
tff(fact_7950_finite__well__order__on__ordIso,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( order_well_order_on(A,A5,R3)
       => ( order_well_order_on(A,A5,R6)
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R6)),bNF_Wellorder_ordIso(A,A))) ) ) ) ).

% finite_well_order_on_ordIso
tff(fact_7951_ordLeq__total,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
          | pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% ordLeq_total
tff(fact_7952_ordIso__reflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R3)),bNF_Wellorder_ordIso(A,A))) ) ).

% ordIso_reflexive
tff(fact_7953_ordLeq__reflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R3)),bNF_Wellorder_ordLeq(A,A))) ) ).

% ordLeq_reflexive
tff(fact_7954_ordLeq__Well__order__simp,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
        & order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6) ) ) ).

% ordLeq_Well_order_simp
tff(fact_7955_ordLeq__ordIso__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_Wellorder_ordIso(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_Wellorder_ordLeq(A,C))) ) ) ).

% ordLeq_ordIso_trans
tff(fact_7956_ordIso__ordLeq__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_Wellorder_ordLeq(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_Wellorder_ordLeq(A,C))) ) ) ).

% ordIso_ordLeq_trans
tff(fact_7957_ordLeq__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_Wellorder_ordLeq(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_Wellorder_ordLeq(A,C))) ) ) ).

% ordLeq_transitive
tff(fact_7958_ordIso__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_Wellorder_ordIso(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_Wellorder_ordIso(A,C))) ) ) ).

% ordIso_transitive
tff(fact_7959_ordIso__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B))) ) ).

% ordIso_imp_ordLeq
tff(fact_7960_ordIso__iff__ordLeq,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
    <=> ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
        & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ).

% ordIso_iff_ordLeq
tff(fact_7961_ordIso__symmetric,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordIso(B,A))) ) ).

% ordIso_symmetric
tff(fact_7962_internalize__ordLeq,axiom,
    ! [A: $tType,B: $tType,R6: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),R3)),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [P4: set(product_prod(B,B))] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),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),R3)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),P4)),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P4),R3)),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_ordLeq
tff(fact_7963_not__ordLess__ordIso,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B))) ) ).

% not_ordLess_ordIso
tff(fact_7964_not__ordLess__ordLeq,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => ~ pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordLeq(B,A))) ) ).

% not_ordLess_ordLeq
tff(fact_7965_ordLess__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B))) ) ).

% ordLess_imp_ordLeq
tff(fact_7966_ordIso__ordLess__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_We4044943003108391690rdLess(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordIso_ordLess_trans
tff(fact_7967_ordLeq__ordLess__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_We4044943003108391690rdLess(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLeq_ordLess_trans
tff(fact_7968_ordLess__ordIso__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_Wellorder_ordIso(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLess_ordIso_trans
tff(fact_7969_ordLess__ordLeq__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),R11: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R6),R11)),bNF_Wellorder_ordLeq(B,C)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R3),R11)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLess_ordLeq_trans
tff(fact_7970_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
    <=> ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
        | pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B))) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
tff(fact_7971_internalize__ordLess,axiom,
    ! [A: $tType,B: $tType,R6: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),R3)),bNF_We4044943003108391690rdLess(A,B)))
    <=> ? [P4: set(product_prod(B,B))] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),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),R3)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),P4)),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P4),R3)),bNF_We4044943003108391690rdLess(B,B))) ) ) ).

% internalize_ordLess
tff(fact_7972_ordLess__or__ordLeq,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
          | pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% ordLess_or_ordLeq
tff(fact_7973_not__ordLeq__iff__ordLess,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( ~ pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordLeq(B,A)))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ).

% not_ordLeq_iff_ordLess
tff(fact_7974_not__ordLess__iff__ordLeq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( ~ pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_We4044943003108391690rdLess(B,A)))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% not_ordLess_iff_ordLeq
tff(fact_7975_ordLeq__def,axiom,
    ! [A2: $tType,A: $tType] : bNF_Wellorder_ordLeq(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_are(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordLeq_def
tff(fact_7976_ordIso__def,axiom,
    ! [A2: $tType,A: $tType] : bNF_Wellorder_ordIso(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_arf(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordIso_def
tff(fact_7977_ofilter__subset__ordLeq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( order_ofilter(A,R3,A5)
       => ( order_ofilter(A,R3,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
          <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B4,aTP_Lamp_yj(set(A),fun(A,set(A)),B4))))),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% ofilter_subset_ordLeq
tff(fact_7978_comp__set__bd__Union__o__collect,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: C,X6: set(fun(C,set(set(A)))),Hbd: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(fun(C,set(set(A)))),set(set(set(A))),image2(fun(C,set(set(A))),set(set(A)),aTP_Lamp_arg(C,fun(fun(C,set(set(A))),set(set(A))),X)),X6))))),Hbd)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A))),bNF_collect(C,set(A),X6)),X))),Hbd)),bNF_Wellorder_ordLeq(A,B))) ) ).

% comp_set_bd_Union_o_collect
tff(fact_7979_dir__image__ordIso,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( inj_on(A,B,F3,aa(set(product_prod(A,A)),set(A),field2(A),R3))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_We2720479622203943262_image(A,B,R3,F3))),bNF_Wellorder_ordIso(A,B))) ) ) ).

% dir_image_ordIso
tff(fact_7980_card__of__least,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,A5,R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_least
tff(fact_7981_card__of__Times__infinite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ( B4 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,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,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(product_prod(A,B),A)))
            & pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,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,B4,aTP_Lamp_mi(set(A),fun(B,set(A)),A5)))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ).

% card_of_Times_infinite
tff(fact_7982_card__of__Times__infinite__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ( B4 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,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,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).

% card_of_Times_infinite_simps(1)
tff(fact_7983_card__of__Times__infinite__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ( B4 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,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,A5)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordIso(A,product_prod(A,B)))) ) ) ) ).

% card_of_Times_infinite_simps(2)
tff(fact_7984_card__of__Times__infinite__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ( B4 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,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,B4,aTP_Lamp_mi(set(A),fun(B,set(A)),A5)))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ).

% card_of_Times_infinite_simps(3)
tff(fact_7985_card__of__Times__infinite__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ( B4 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,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,A5)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B4,aTP_Lamp_mi(set(A),fun(B,set(A)),A5))))),bNF_Wellorder_ordIso(A,product_prod(B,A)))) ) ) ) ).

% card_of_Times_infinite_simps(4)
tff(fact_7986_card__of__Times1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,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,B4)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B4,aTP_Lamp_mi(set(A),fun(B,set(A)),A5))))),bNF_Wellorder_ordLeq(B,product_prod(B,A)))) ) ).

% card_of_Times1
tff(fact_7987_card__of__Times2,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,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,B4)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ).

% card_of_Times2
tff(fact_7988_card__of__bool,axiom,
    ! [A: $tType,A1: A,A22: A] :
      ( ( A1 != A22 )
     => pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(bool,top_top(set(bool)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A22),bot_bot(set(A))))))),bNF_Wellorder_ordIso(bool,A))) ) ).

% card_of_bool
tff(fact_7989_card__of__empty,axiom,
    ! [B: $tType,A: $tType,A5: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordLeq(A,B))) ).

% card_of_empty
tff(fact_7990_card__of__empty2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,B)))
     => ( A5 = bot_bot(set(A)) ) ) ).

% card_of_empty2
tff(fact_7991_card__of__empty3,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordLeq(A,B)))
     => ( A5 = bot_bot(set(A)) ) ) ).

% card_of_empty3
tff(fact_7992_card__of__empty__ordIso,axiom,
    ! [B: $tType,A: $tType] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),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_7993_card__of__ordIso,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ? [F10: fun(A,B)] : bij_betw(A,B,F10,A5,B4)
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_ordIso
tff(fact_7994_card__of__ordIsoI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F3,A5,B4)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_ordIsoI
tff(fact_7995_ex__bij__betw,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ? [F2: fun(B,A),B10: set(B)] : bij_betw(B,A,F2,B10,A5) ) ).

% ex_bij_betw
tff(fact_7996_card__of__Sigma__ordLeq__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: set(A),I5: set(B),A5: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordLeq(B,A)))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),I5))
             => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A5,X3))),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordLeq(C,A))) )
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A5))),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ).

% card_of_Sigma_ordLeq_infinite
tff(fact_7997_card__of__Times__same__infinite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5)))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(product_prod(A,A),A))) ) ).

% card_of_Times_same_infinite
tff(fact_7998_card__of__Times3,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A5,aTP_Lamp_yj(set(A),fun(A,set(A)),A5))))),bNF_Wellorder_ordLeq(A,product_prod(A,A)))) ).

% card_of_Times3
tff(fact_7999_card__of__Sigma__mono1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),B4: fun(A,set(C))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),I5))
         => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A5,X3))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),B4,X3)))),bNF_Wellorder_ordLeq(B,C))) )
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I5,A5))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,I5,B4)))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C)))) ) ).

% card_of_Sigma_mono1
tff(fact_8000_card__of__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,A5,aTP_Lamp_yh(set(C),fun(A,set(C)),C4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,B4,aTP_Lamp_yg(set(C),fun(B,set(C)),C4))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% card_of_Times_mono1
tff(fact_8001_card__of__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,C4,aTP_Lamp_arh(set(A),fun(C,set(A)),A5)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,C4,aTP_Lamp_ari(set(B),fun(C,set(B)),B4))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% card_of_Times_mono2
tff(fact_8002_card__of__Times__commute,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] : pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,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,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B4,aTP_Lamp_mi(set(A),fun(B,set(A)),A5))))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).

% card_of_Times_commute
tff(fact_8003_card__of__refl,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(A,A))) ).

% card_of_refl
tff(fact_8004_card__of__ordLeqI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B4: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( ! [A6: A] :
            ( pp(aa(set(A),bool,member(A,A6),A5))
           => pp(aa(set(B),bool,member(B,aa(A,B,F3,A6)),B4)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B))) ) ) ).

% card_of_ordLeqI
tff(fact_8005_infinite__iff__card__of__nat,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
    <=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(nat,A))) ) ).

% infinite_iff_card_of_nat
tff(fact_8006_card__of__ordIso__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(A),bool,finite_finite2(A),A5))
      <=> pp(aa(set(B),bool,finite_finite2(B),B4)) ) ) ).

% card_of_ordIso_finite
tff(fact_8007_card__of__ordLeq__finite,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).

% card_of_ordLeq_finite
tff(fact_8008_card__of__ordLeq__infinite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
       => ~ pp(aa(set(B),bool,finite_finite2(B),B4)) ) ) ).

% card_of_ordLeq_infinite
tff(fact_8009_card__of__cong,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3))),bNF_Ca6860139660246222851ard_of(B,aa(set(product_prod(B,B)),set(B),field2(B),R6)))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_cong
tff(fact_8010_card__of__mono2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3))),bNF_Ca6860139660246222851ard_of(B,aa(set(product_prod(B,B)),set(B),field2(B),R6)))),bNF_Wellorder_ordLeq(A,B))) ) ).

% card_of_mono2
tff(fact_8011_card__of__UNION__Sigma,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),I5: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,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),A5),I5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,I5,A5)))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ).

% card_of_UNION_Sigma
tff(fact_8012_card__of__image,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(B),set(A),image2(B,A,F3),A5))),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordLeq(A,B))) ).

% card_of_image
tff(fact_8013_type__copy__set__bd,axiom,
    ! [A: $tType,D: $tType,C: $tType,B: $tType,S: fun(A,set(B)),Bd: set(product_prod(C,C)),Rep: fun(D,A),X: D] :
      ( ! [Y3: A] : pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),S,Y3))),Bd)),bNF_Wellorder_ordLeq(B,C)))
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),aa(fun(D,A),fun(D,set(B)),comp(A,set(B),D,S),Rep),X))),Bd)),bNF_Wellorder_ordLeq(B,C))) ) ).

% type_copy_set_bd
tff(fact_8014_card__of__Pow__Func,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),bNF_Ca6860139660246222851ard_of(set(A),pow(A,A5))),bNF_Ca6860139660246222851ard_of(fun(A,bool),bNF_Wellorder_Func(A,bool,A5,top_top(set(bool)))))),bNF_Wellorder_ordIso(set(A),fun(A,bool)))) ).

% card_of_Pow_Func
tff(fact_8015_Func__Times__Range,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),bool,member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Ca6860139660246222851ard_of(fun(A,product_prod(B,C)),bNF_Wellorder_Func(A,product_prod(B,C),A5,product_Sigma(B,C,B4,aTP_Lamp_yg(set(C),fun(B,set(C)),C4))))),bNF_Ca6860139660246222851ard_of(product_prod(fun(A,B),fun(A,C)),product_Sigma(fun(A,B),fun(A,C),bNF_Wellorder_Func(A,B,A5,B4),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_arj(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),A5),C4))))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).

% Func_Times_Range
tff(fact_8016_card__of__Func__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),bool,member(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),fun(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),product_Pair(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),bNF_Ca6860139660246222851ard_of(fun(product_prod(A,B),C),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),C4))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,C)),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C4))))),bNF_Wellorder_ordIso(fun(product_prod(A,B),C),fun(A,fun(B,C))))) ).

% card_of_Func_Times
tff(fact_8017_card__of__mono1,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_mono1
tff(fact_8018_card__of__Pow,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(set(A),pow(A,A5)))),bNF_We4044943003108391690rdLess(A,set(A)))) ).

% card_of_Pow
tff(fact_8019_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3))),R6)),bNF_We4044943003108391690rdLess(A,B))) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_8020_dir__image__def,axiom,
    ! [A2: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,A2)] : bNF_We2720479622203943262_image(A,A2,R3,F3) = aa(fun(product_prod(A2,A2),bool),set(product_prod(A2,A2)),collect(product_prod(A2,A2)),aa(fun(A,A2),fun(product_prod(A2,A2),bool),aTP_Lamp_ark(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),R3),F3)) ).

% dir_image_def
tff(fact_8021_card__of__ordLeq2,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ? [G7: fun(B,A)] : aa(set(B),set(A),image2(B,A,G7),B4) = A5
      <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B))) ) ) ).

% card_of_ordLeq2
tff(fact_8022_surj__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,B4: set(A),F3: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(set(B),set(A),image2(B,A,F3),A5)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B4)),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordLeq(A,B))) ) ).

% surj_imp_ordLeq
tff(fact_8023_card__of__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B2: B] :
      ( ( A5 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_singl_ordLeq
tff(fact_8024_card__of__ordLess2,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A5: set(B)] :
      ( ( B4 != bot_bot(set(A)) )
     => ( ~ ? [F10: fun(B,A)] : aa(set(B),set(A),image2(B,A,F10),A5) = B4
      <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A5)),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_We4044943003108391690rdLess(B,A))) ) ) ).

% card_of_ordLess2
tff(fact_8025_internalize__card__of__ordLeq2,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C4: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,C4))),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B7: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),C4))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B7))),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B7)),bNF_Ca6860139660246222851ard_of(B,C4))),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq2
tff(fact_8026_card__of__Field__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3))),R3)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_Field_ordLess
tff(fact_8027_card__of__Func__UNIV,axiom,
    ! [B: $tType,A: $tType,B4: set(B)] : pp(aa(set(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),bool,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)),B4))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_arl(set(B),fun(fun(A,B),bool),B4))))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B)))) ).

% card_of_Func_UNIV
tff(fact_8028_ordLeq__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),A5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,A5,aTP_Lamp_arm(set(product_prod(A,A)),fun(C,set(A)),R3)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,A5,aTP_Lamp_arn(set(product_prod(B,B)),fun(C,set(B)),R6))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% ordLeq_Times_mono2
tff(fact_8029_ordLeq__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_yh(set(C),fun(A,set(C)),C4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,aa(set(product_prod(B,B)),set(B),field2(B),R6),aTP_Lamp_yg(set(C),fun(B,set(C)),C4))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% ordLeq_Times_mono1
tff(fact_8030_card__of__ordLeq,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ? [F10: fun(A,B)] :
          ( inj_on(A,B,F10,A5)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F10),A5)),B4)) )
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B))) ) ).

% card_of_ordLeq
tff(fact_8031_internalize__card__of__ordLeq,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B7: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),aa(set(product_prod(B,B)),set(B),field2(B),R3)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B7))),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B7)),R3)),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq
tff(fact_8032_card__of__ordLess,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ~ ? [F10: fun(A,B)] :
            ( inj_on(A,B,F10,A5)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F10),A5)),B4)) )
    <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_We4044943003108391690rdLess(B,A))) ) ).

% card_of_ordLess
tff(fact_8033_card__of__UNION__ordLeq__infinite,axiom,
    ! [B: $tType,A: $tType,C: $tType,B4: set(A),I5: set(B),A5: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordLeq(B,A)))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),I5))
             => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A5,X3))),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordLeq(C,A))) )
         => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A5),I5)))),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordLeq(C,A))) ) ) ) ).

% card_of_UNION_ordLeq_infinite
tff(fact_8034_regularCard__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca7133664381575040944arCard(A,R3)
    <=> ! [K9: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K9),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
            & bNF_Ca7293521722713021262ofinal(A,K9,R3) )
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,K9)),R3)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% regularCard_def
tff(fact_8035_card__of__ordIso__subst,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( ( A5 = B4 )
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(A,B4))),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_ordIso_subst
tff(fact_8036_cone__def,axiom,
    bNF_Cardinal_cone = bNF_Ca6860139660246222851ard_of(product_unit,aa(set(product_unit),set(product_unit),aa(product_unit,fun(set(product_unit),set(product_unit)),insert(product_unit),product_Unity),bot_bot(set(product_unit)))) ).

% cone_def
tff(fact_8037_SIGMA__CSUM,axiom,
    ! [B: $tType,A: $tType,I5: set(A),As9: fun(A,set(B))] : bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I5,As9)) = bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I5),aTP_Lamp_aro(fun(A,set(B)),fun(A,set(product_prod(B,B))),As9)) ).

% SIGMA_CSUM
tff(fact_8038_Csum__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),Rs: fun(A,set(product_prod(B,B)))] : bNF_Cardinal_Csum(A,B,R3,Rs) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_arp(fun(A,set(product_prod(B,B))),fun(A,set(B)),Rs))) ).

% Csum_def
tff(fact_8039_card__of__Times__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),A5: set(B),B4: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A5)),R3)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B4)),R3)),bNF_Wellorder_ordLeq(C,A)))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,A5,aTP_Lamp_yg(set(C),fun(B,set(C)),B4)))),R3)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Times_ordLeq_infinite_Field
tff(fact_8040_card__of__Sigma__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R3)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A5,X3))),R3)),bNF_Wellorder_ordLeq(C,A))) )
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A5))),R3)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Sigma_ordLeq_infinite_Field
tff(fact_8041_card__of__unique,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,A5,R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_unique
tff(fact_8042_Card__order__wo__rel,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => bNF_Wellorder_wo_rel(A,R3) ) ).

% Card_order_wo_rel
tff(fact_8043_card__order__on__def,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,A5,R3)
    <=> ( order_well_order_on(A,A5,R3)
        & ! [R14: set(product_prod(A,A))] :
            ( order_well_order_on(A,A5,R14)
           => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R14)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ).

% card_order_on_def
tff(fact_8044_card__order__on__ordIso,axiom,
    ! [A: $tType,A5: set(A),R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,A5,R3)
     => ( bNF_Ca8970107618336181345der_on(A,A5,R6)
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R6)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% card_order_on_ordIso
tff(fact_8045_Card__order__ordIso,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordIso(B,A)))
       => bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6) ) ) ).

% Card_order_ordIso
tff(fact_8046_Card__order__ordIso2,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
       => bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6) ) ) ).

% Card_order_ordIso2
tff(fact_8047_ordLeq__refl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R3)),bNF_Wellorder_ordLeq(A,A))) ) ).

% ordLeq_refl
tff(fact_8048_ordIso__refl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R3)),bNF_Wellorder_ordIso(A,A))) ) ).

% ordIso_refl
tff(fact_8049_Card__order__trans,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ( X != Y )
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
         => ( ( Y != Z2 )
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R3))
             => ( ( X != Z2 )
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R3)) ) ) ) ) ) ) ).

% Card_order_trans
tff(fact_8050_infinite__Card__order__limit,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
              & ( A3 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),R3)) ) ) ) ) ).

% infinite_Card_order_limit
tff(fact_8051_Card__order__iff__ordLeq__card__of,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3)))),bNF_Wellorder_ordLeq(A,A))) ) ).

% Card_order_iff_ordLeq_card_of
tff(fact_8052_ordIso__card__of__imp__Card__order,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A5: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordIso(A,B)))
     => bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) ) ).

% ordIso_card_of_imp_Card_order
tff(fact_8053_Card__order__iff__ordIso__card__of,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3)))),bNF_Wellorder_ordIso(A,A))) ) ).

% Card_order_iff_ordIso_card_of
tff(fact_8054_card__of__Field__ordIso,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3))),R3)),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_Field_ordIso
tff(fact_8055_dir__image,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),R3: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A] :
          ( ( aa(A,B,F3,X3) = aa(A,B,F3,Y3) )
        <=> ( X3 = Y3 ) )
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_We2720479622203943262_image(A,B,R3,F3))),bNF_Wellorder_ordIso(A,B))) ) ) ).

% dir_image
tff(fact_8056_exists__minim__Card__order,axiom,
    ! [A: $tType,R2: set(set(product_prod(A,A)))] :
      ( ( R2 != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X3),R2))
           => bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),X3),X3) )
       => ? [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X3),R2))
            & ! [Xa3: set(product_prod(A,A))] :
                ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),Xa3),R2))
               => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% exists_minim_Card_order
tff(fact_8057_Card__order__empty,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ).

% Card_order_empty
tff(fact_8058_card__of__ordIso__finite__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),A5: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordIso(A,B)))
       => ( pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
        <=> pp(aa(set(B),bool,finite_finite2(B),A5)) ) ) ) ).

% card_of_ordIso_finite_Field
tff(fact_8059_card__of__underS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,order_underS(A,R3,A3))),R3)),bNF_We4044943003108391690rdLess(A,A))) ) ) ).

% card_of_underS
tff(fact_8060_Card__order__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),B2: B] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R3) != bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B2),bot_bot(set(B))))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ).

% Card_order_singl_ordLeq
tff(fact_8061_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),A5: set(B),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A5)),R3)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),R3)),bNF_Wellorder_ordLeq(B,A)))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
           => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B4))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
tff(fact_8062_card__of__empty1,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A))] :
      ( ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
        | bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_empty1
tff(fact_8063_Card__order__Pow,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),bNF_Ca6860139660246222851ard_of(set(A),pow(A,aa(set(product_prod(A,A)),set(A),field2(A),R3))))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% Card_order_Pow
tff(fact_8064_Card__order__Times1,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),B4: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ( B4 != bot_bot(set(B)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,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)))),R3),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordLeq(A,product_prod(A,B)))) ) ) ).

% Card_order_Times1
tff(fact_8065_Card__order__Times2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A5: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ( A5 != bot_bot(set(B)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,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)))),R3),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A5,aTP_Lamp_arq(set(product_prod(A,A)),fun(B,set(A)),R3))))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ) ) ).

% Card_order_Times2
tff(fact_8066_Card__order__Times__same__infinite,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_aje(set(product_prod(A,A)),fun(A,set(A)),R3)))),R3)),bNF_Wellorder_ordLeq(product_prod(A,A),A))) ) ) ).

% Card_order_Times_same_infinite
tff(fact_8067_card__of__UNION__ordLeq__infinite__Field,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R3)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A5,X3))),R3)),bNF_Wellorder_ordLeq(C,A))) )
           => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A5),I5)))),R3)),bNF_Wellorder_ordLeq(C,A))) ) ) ) ) ).

% card_of_UNION_ordLeq_infinite_Field
tff(fact_8068_Card__order__iff__Restr__underS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
           => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_arb(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4)))),bNF_Ca6860139660246222851ard_of(A,aa(set(product_prod(A,A)),set(A),field2(A),R3)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).

% Card_order_iff_Restr_underS
tff(fact_8069_regularCard__UNION,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),As9: fun(A,set(B)),B4: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca7133664381575040944arCard(A,R3)
       => ( bNF_Ca3754400796208372196lChain(A,set(B),R3,As9)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As9),aa(set(product_prod(A,A)),set(A),field2(A),R3)))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),R3)),bNF_We4044943003108391690rdLess(B,A)))
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(A,set(B),As9,X3))) ) ) ) ) ) ) ).

% regularCard_UNION
tff(fact_8070_Card__order__Times__infinite,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
       => ( ( aa(set(product_prod(B,B)),set(B),field2(B),P3) != bot_bot(set(B)) )
         => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R3)),bNF_Wellorder_ordLeq(B,A)))
           => ( pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,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),R3),aTP_Lamp_arr(set(product_prod(B,B)),fun(A,set(B)),P3)))),R3)),bNF_Wellorder_ordIso(product_prod(A,B),A)))
              & pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,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),P3),aTP_Lamp_arq(set(product_prod(A,A)),fun(B,set(A)),R3)))),R3)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ) ).

% Card_order_Times_infinite
tff(fact_8071_ex__toCard__pred,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3)
       => ? [X_1: fun(A,B)] : bNF_Gr1419584066657907630d_pred(A,B,A5,R3,X_1) ) ) ).

% ex_toCard_pred
tff(fact_8072_toCard__pred__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R3: set(product_prod(B,B)),F3: fun(A,B)] :
      ( bNF_Gr1419584066657907630d_pred(A,B,A5,R3,F3)
    <=> ( inj_on(A,B,F3,A5)
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),aa(set(product_prod(B,B)),set(B),field2(B),R3)))
        & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3) ) ) ).

% toCard_pred_def
tff(fact_8073_toCard__pred__toCard,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3)
       => bNF_Gr1419584066657907630d_pred(A,B,A5,R3,bNF_Greatest_toCard(A,B,A5,R3)) ) ) ).

% toCard_pred_toCard
tff(fact_8074_cardSuc__UNION,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),As9: fun(set(A),set(B)),B4: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R3),As9)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As9),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R3))))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),R3)),bNF_Wellorder_ordLeq(B,A)))
             => ? [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X3),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R3))))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),As9,X3))) ) ) ) ) ) ) ).

% cardSuc_UNION
tff(fact_8075_cardSuc__ordLeq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),bNF_Ca8387033319878233205ardSuc(A,R3))),bNF_Wellorder_ordLeq(A,set(A)))) ) ).

% cardSuc_ordLeq
tff(fact_8076_cardSuc__greater,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),bNF_Ca8387033319878233205ardSuc(A,R3))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% cardSuc_greater
tff(fact_8077_cardSuc__ordLess__ordLeq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
        <=> pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R3)),R6)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_ordLess_ordLeq
tff(fact_8078_cardSuc__least,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
         => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R3)),R6)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_least
tff(fact_8079_cardSuc__mono__ordLeq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R3)),bNF_Ca8387033319878233205ardSuc(B,R6))),bNF_Wellorder_ordLeq(set(A),set(B))))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% cardSuc_mono_ordLeq
tff(fact_8080_cardSuc__invar__ordIso,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R3)),bNF_Ca8387033319878233205ardSuc(B,R6))),bNF_Wellorder_ordIso(set(A),set(B))))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B))) ) ) ) ).

% cardSuc_invar_ordIso
tff(fact_8081_cardSuc__least__aux,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(set(A),set(A)))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca8970107618336181345der_on(set(A),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),R6)),bNF_We4044943003108391690rdLess(A,set(A))))
         => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool,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,R3)),R6)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% cardSuc_least_aux
tff(fact_8082_cardSuc__ordLeq__ordLess,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(B,B)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),R6),bNF_Ca8387033319878233205ardSuc(A,R3))),bNF_We4044943003108391690rdLess(B,set(A))))
        <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% cardSuc_ordLeq_ordLess
tff(fact_8083_toCard__inj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(B,B)),X: A,Y: A] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3)
       => ( pp(aa(set(A),bool,member(A,X),A5))
         => ( pp(aa(set(A),bool,member(A,Y),A5))
           => ( ( aa(A,B,bNF_Greatest_toCard(A,B,A5,R3),X) = aa(A,B,bNF_Greatest_toCard(A,B,A5,R3),Y) )
            <=> ( X = Y ) ) ) ) ) ) ).

% toCard_inj
tff(fact_8084_fromCard__toCard,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(B,B)),B2: A] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3)
       => ( pp(aa(set(A),bool,member(A,B2),A5))
         => ( bNF_Gr5436034075474128252omCard(A,B,A5,R3,aa(A,B,bNF_Greatest_toCard(A,B,A5,R3),B2)) = B2 ) ) ) ) ).

% fromCard_toCard
tff(fact_8085_isCardSuc__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(set(A),set(A)))] :
      ( bNF_Ca6246979054910435723ardSuc(A,R3,R6)
    <=> ( bNF_Ca8970107618336181345der_on(set(A),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),R6),R6)
        & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),R6)),bNF_We4044943003108391690rdLess(A,set(A))))
        & ! [R15: 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)),R15),R15)
              & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),R15)),bNF_We4044943003108391690rdLess(A,set(A)))) )
           => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool,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)))),R6),R15)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% isCardSuc_def
tff(fact_8086_cardSuc__UNION__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),As9: fun(set(A),set(B)),B4: set(B)] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R3),As9)
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As9),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R3))))))
         => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),R3)),bNF_Wellorder_ordLeq(B,A)))
           => ? [X3: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),X3),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R3))))
                & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),As9,X3))) ) ) ) ) ) ).

% cardSuc_UNION_Cinfinite
tff(fact_8087_card__of__Csum__Times_H,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(C))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),I5))
           => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A5,X3))),R3)),bNF_Wellorder_ordLeq(C,A))) )
       => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_Csum(B,C,bNF_Ca6860139660246222851ard_of(B,I5),aTP_Lamp_ars(fun(B,set(C)),fun(B,set(product_prod(C,C))),A5))),bNF_Cardinal_cprod(B,A,bNF_Ca6860139660246222851ard_of(B,I5),R3))),bNF_Wellorder_ordLeq(product_prod(B,C),product_prod(B,A)))) ) ) ).

% card_of_Csum_Times'
tff(fact_8088_Cinfinite__limit2,axiom,
    ! [A: $tType,X1: A,R3: set(product_prod(A,A)),X2: A] :
      ( pp(aa(set(A),bool,member(A,X1),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( pp(aa(set(A),bool,member(A,X2),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( bNF_Ca4139267488887388095finite(A,R3)
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
              & ( X1 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X3)),R3))
              & ( X2 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X3)),R3)) ) ) ) ) ).

% Cinfinite_limit2
tff(fact_8089_Cinfinite__limit,axiom,
    ! [A: $tType,X: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
     => ( ( bNF_Ca4139267488887388095finite(A,R3)
          & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
            & ( X != X3 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X3)),R3)) ) ) ) ).

% Cinfinite_limit
tff(fact_8090_cprod__infinite,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Cardinal_cprod(A,A,R3,R3)),R3)),bNF_Wellorder_ordIso(product_prod(A,A),A))) ) ).

% cprod_infinite
tff(fact_8091_cinfinite__mono,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R12),R23)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca4139267488887388095finite(A,R12)
       => bNF_Ca4139267488887388095finite(B,R23) ) ) ).

% cinfinite_mono
tff(fact_8092_cprod__com,axiom,
    ! [B: $tType,A: $tType,P13: set(product_prod(A,A)),P25: set(product_prod(B,B))] : pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,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,P25)),bNF_Cardinal_cprod(B,A,P25,P13))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).

% cprod_com
tff(fact_8093_cprod__cinfinite__bound,axiom,
    ! [B: $tType,C: $tType,A: $tType,P3: set(product_prod(A,A)),R3: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P3),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),Q3),R3)),bNF_Wellorder_ordLeq(C,B)))
       => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P3),P3)
         => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
           => ( ( bNF_Ca4139267488887388095finite(B,R3)
                & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3) )
             => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,C,P3,Q3)),R3)),bNF_Wellorder_ordLeq(product_prod(A,C),B))) ) ) ) ) ) ).

% cprod_cinfinite_bound
tff(fact_8094_cprod__dup,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(B,B)),P10: set(product_prod(C,C))] :
      ( bNF_Ca4139267488887388095finite(A,R3)
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
       => ( pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Cardinal_cprod(B,C,P3,P10)),bNF_Cardinal_cprod(A,A,R3,R3))),bNF_Wellorder_ordIso(product_prod(B,C),product_prod(A,A))))
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Cardinal_cprod(B,C,P3,P10)),R3)),bNF_Wellorder_ordIso(product_prod(B,C),A))) ) ) ) ).

% cprod_dup
tff(fact_8095_cprod__def,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] : bNF_Cardinal_cprod(A,B,R12,R23) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R12),aTP_Lamp_arr(set(product_prod(B,B)),fun(A,set(B)),R23))) ).

% cprod_def
tff(fact_8096_Cinfinite__cong,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),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_8097_Cinfinite__limit__finite,axiom,
    ! [A: $tType,X6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
       => ( ( bNF_Ca4139267488887388095finite(A,R3)
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(set(product_prod(A,A)),set(A),field2(A),R3)))
              & ! [Xa3: A] :
                  ( pp(aa(set(A),bool,member(A,Xa3),X6))
                 => ( ( Xa3 != X3 )
                    & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3)),R3)) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
tff(fact_8098_cprod__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P25: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P25),R23)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Cardinal_cprod(C,A,Q3,P25)),bNF_Cardinal_cprod(C,B,Q3,R23))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% cprod_mono2
tff(fact_8099_cprod__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Cardinal_cprod(A,C,P13,Q3)),bNF_Cardinal_cprod(B,C,R12,Q3))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% cprod_mono1
tff(fact_8100_cprod__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P25: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P25),R23)),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P13,P25)),bNF_Cardinal_cprod(B,D,R12,R23))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,D)))) ) ) ).

% cprod_mono
tff(fact_8101_cprod__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P25: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P25),R23)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Cardinal_cprod(C,A,Q3,P25)),bNF_Cardinal_cprod(C,B,Q3,R23))),bNF_Wellorder_ordIso(product_prod(C,A),product_prod(C,B)))) ) ).

% cprod_cong2
tff(fact_8102_cprod__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P25: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Cardinal_cprod(A,C,P13,P25)),bNF_Cardinal_cprod(B,C,R12,P25))),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,C)))) ) ).

% cprod_cong1
tff(fact_8103_cprod__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P25: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P25),R23)),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P13,P25)),bNF_Cardinal_cprod(B,D,R12,R23))),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,D)))) ) ) ).

% cprod_cong
tff(fact_8104_Un__Cinfinite__bound,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R3: set(product_prod(B,B)),B4: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B4)),R3)),bNF_Wellorder_ordLeq(A,B)))
       => ( ( bNF_Ca4139267488887388095finite(B,R3)
            & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3) )
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))),R3)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% Un_Cinfinite_bound
tff(fact_8105_UNION__Cinfinite__bound,axiom,
    ! [A: $tType,B: $tType,C: $tType,I5: set(A),R3: set(product_prod(B,B)),A5: fun(A,set(C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,I5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),I5))
           => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),A5,X3))),R3)),bNF_Wellorder_ordLeq(C,B))) )
       => ( ( bNF_Ca4139267488887388095finite(B,R3)
            & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3) )
         => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A5),I5)))),R3)),bNF_Wellorder_ordLeq(C,B))) ) ) ) ).

% UNION_Cinfinite_bound
tff(fact_8106_card__of__Csum__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),B4: set(C)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),I5))
         => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A5,X3))),bNF_Ca6860139660246222851ard_of(C,B4))),bNF_Wellorder_ordLeq(B,C))) )
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I5),aTP_Lamp_aro(fun(A,set(B)),fun(A,set(product_prod(B,B))),A5))),bNF_Cardinal_cprod(A,C,bNF_Ca6860139660246222851ard_of(A,I5),bNF_Ca6860139660246222851ard_of(C,B4)))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C)))) ) ).

% card_of_Csum_Times
tff(fact_8107_comp__single__set__bd,axiom,
    ! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set(product_prod(A,A)),Fset: fun(B,set(C)),Gset: fun(D,set(B)),Gbd: set(product_prod(E,E)),X: D] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Fbd),Fbd)
     => ( ! [X3: B] : pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Fset,X3))),Fbd)),bNF_Wellorder_ordLeq(C,A)))
       => ( ! [X3: D] : pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),Gset,X3))),Gbd)),bNF_Wellorder_ordLeq(B,E)))
         => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(C,C)),fun(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),product_Pair(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Fset),aa(D,set(B),Gset,X))))),bNF_Cardinal_cprod(E,A,Gbd,Fbd))),bNF_Wellorder_ordLeq(C,product_prod(E,A)))) ) ) ) ).

% comp_single_set_bd
tff(fact_8108_Cfinite__cprod__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( ( bNF_Ca4139267488887388095finite(B,S2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),S2),S2) )
       => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B)))),bool,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,R3,S2)),S2)),bNF_Wellorder_ordLeq(product_prod(A,B),B))) ) ) ).

% Cfinite_cprod_Cinfinite
tff(fact_8109_cprod__infinite1_H,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( ( ~ pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P3),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(B,B)))
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),P3),P3) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R3)),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,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,R3,P3)),R3)),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).

% cprod_infinite1'
tff(fact_8110_czero__def,axiom,
    ! [A: $tType] : bNF_Cardinal_czero(A) = bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A))) ).

% czero_def
tff(fact_8111_cone__not__czero,axiom,
    ! [A: $tType] : ~ pp(aa(set(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),bool,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_8112_czero__ordIso,axiom,
    ! [B: $tType,A: $tType] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Cardinal_czero(A)),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B))) ).

% czero_ordIso
tff(fact_8113_cinfinite__not__czero,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,B))] :
      ( bNF_Ca4139267488887388095finite(B,R3)
     => ~ pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(B,A))) ) ).

% cinfinite_not_czero
tff(fact_8114_czeroE,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B)))
     => ( aa(set(product_prod(A,A)),set(A),field2(A),R3) = bot_bot(set(A)) ) ) ).

% czeroE
tff(fact_8115_card__of__ordIso__czero__iff__empty,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B)))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% card_of_ordIso_czero_iff_empty
tff(fact_8116_czeroI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R3) = bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B))) ) ) ).

% czeroI
tff(fact_8117_Cnotzero__imp__not__empty,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( aa(set(product_prod(A,A)),set(A),field2(A),R3) != bot_bot(set(A)) ) ) ).

% Cnotzero_imp_not_empty
tff(fact_8118_Cnotzero__mono,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
      ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),Q3),Q3)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),Q3)),bNF_Wellorder_ordLeq(A,B)))
         => ( ~ pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),Q3),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(B,B)))
            & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),Q3),Q3) ) ) ) ) ).

% Cnotzero_mono
tff(fact_8119_Cinfinite__Cnotzero,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) ) ) ).

% Cinfinite_Cnotzero
tff(fact_8120_cone__ordLeq__Cnotzero,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => pp(aa(set(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),bool,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),R3)),bNF_Wellorder_ordLeq(product_unit,A))) ) ).

% cone_ordLeq_Cnotzero
tff(fact_8121_Cnotzero__UNIV,axiom,
    ! [A: $tType] :
      ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,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_8122_cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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_8123_ordLeq__cprod2,axiom,
    ! [A: $tType,B: $tType,P13: set(product_prod(A,A)),P25: set(product_prod(B,B))] :
      ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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),P25),P25)
       => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),P25),bNF_Cardinal_cprod(A,B,P13,P25))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ) ).

% ordLeq_cprod2
tff(fact_8124_Cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] :
      ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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_8125_Cfinite__ordLess__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( ( bNF_Ca4139267488887388095finite(B,S2)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),S2),S2) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),S2)),bNF_We4044943003108391690rdLess(A,B))) ) ) ).

% Cfinite_ordLess_Cinfinite
tff(fact_8126_cexp__mono,axiom,
    ! [E: $tType,F: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P25: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P25),R23)),bNF_Wellorder_ordLeq(C,D)))
       => ( ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(E,E)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(C,C)),fun(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E)))),product_Pair(set(product_prod(C,C)),set(product_prod(E,E))),P25),bNF_Cardinal_czero(E))),bNF_Wellorder_ordIso(C,E)))
           => pp(aa(set(product_prod(set(product_prod(D,D)),set(product_prod(F,F)))),bool,member(product_prod(set(product_prod(D,D)),set(product_prod(F,F))),aa(set(product_prod(F,F)),product_prod(set(product_prod(D,D)),set(product_prod(F,F))),aa(set(product_prod(D,D)),fun(set(product_prod(F,F)),product_prod(set(product_prod(D,D)),set(product_prod(F,F)))),product_Pair(set(product_prod(D,D)),set(product_prod(F,F))),R23),bNF_Cardinal_czero(F))),bNF_Wellorder_ordIso(D,F))) )
         => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),P25),P25)
           => pp(aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),bool,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P13,P25)),bNF_Cardinal_cexp(B,D,R12,R23))),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B)))) ) ) ) ) ).

% cexp_mono
tff(fact_8127_cexp__mono2,axiom,
    ! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P25: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P25),R23)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
       => ( ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(A,A)),fun(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D)))),product_Pair(set(product_prod(A,A)),set(product_prod(D,D))),P25),bNF_Cardinal_czero(D))),bNF_Wellorder_ordIso(A,D)))
           => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),R23),bNF_Cardinal_czero(E))),bNF_Wellorder_ordIso(B,E))) )
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P25),P25)
           => pp(aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),bool,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P25)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ) ).

% cexp_mono2
tff(fact_8128_cprod__cexp,axiom,
    ! [C: $tType,B: $tType,A: $tType,R3: set(product_prod(B,B)),S2: set(product_prod(C,C)),T6: set(product_prod(A,A))] : pp(aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),bool,member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R3,S2),T6)),bNF_Cardinal_cprod(fun(A,B),fun(A,C),bNF_Cardinal_cexp(B,A,R3,T6),bNF_Cardinal_cexp(C,A,S2,T6)))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).

% cprod_cexp
tff(fact_8129_cexp__cprod,axiom,
    ! [A: $tType,C: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(C,C)),R32: set(product_prod(B,B))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R12),R12)
     => pp(aa(set(product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))))),bool,member(product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),fun(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))))),product_Pair(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),bNF_Cardinal_cexp(fun(C,A),B,bNF_Cardinal_cexp(A,C,R12,R23),R32)),bNF_Cardinal_cexp(A,product_prod(C,B),R12,bNF_Cardinal_cprod(C,B,R23,R32)))),bNF_Wellorder_ordIso(fun(B,fun(C,A)),fun(product_prod(C,B),A)))) ) ).

% cexp_cprod
tff(fact_8130_cexp__cone,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A)))),bool,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,R3,bNF_Cardinal_cone)),R3)),bNF_Wellorder_ordIso(fun(product_unit,A),A))) ) ).

% cexp_cone
tff(fact_8131_cexp__cprod__ordLeq,axiom,
    ! [A: $tType,B: $tType,C: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),R32: set(product_prod(C,C))] :
      ( 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) )
       => ( ( ~ pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),fun(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C)))),product_Pair(set(product_prod(C,C)),set(product_prod(C,C))),R32),bNF_Cardinal_czero(C))),bNF_Wellorder_ordIso(C,C)))
            & bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),R32),R32) )
         => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),R32),R23)),bNF_Wellorder_ordLeq(C,B)))
           => pp(aa(set(product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A))))),bool,member(product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),bNF_Cardinal_cexp(fun(B,A),C,bNF_Cardinal_cexp(A,B,R12,R23),R32)),bNF_Cardinal_cexp(A,B,R12,R23))),bNF_Wellorder_ordIso(fun(C,fun(B,A)),fun(B,A)))) ) ) ) ) ).

% cexp_cprod_ordLeq
tff(fact_8132_cexp__mono_H,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P25: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P25),R23)),bNF_Wellorder_ordLeq(C,D)))
       => ( ( ( aa(set(product_prod(C,C)),set(C),field2(C),P25) = bot_bot(set(C)) )
           => ( aa(set(product_prod(D,D)),set(D),field2(D),R23) = bot_bot(set(D)) ) )
         => pp(aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),bool,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P13,P25)),bNF_Cardinal_cexp(B,D,R12,R23))),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B)))) ) ) ) ).

% cexp_mono'
tff(fact_8133_cexp__mono1,axiom,
    ! [B: $tType,A: $tType,C: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
       => pp(aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),bool,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P13,Q3)),bNF_Cardinal_cexp(B,C,R12,Q3))),bNF_Wellorder_ordLeq(fun(C,A),fun(C,B)))) ) ) ).

% cexp_mono1
tff(fact_8134_cexp__mono2_H,axiom,
    ! [B: $tType,C: $tType,A: $tType,P25: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P25),R23)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
       => ( ( ( aa(set(product_prod(A,A)),set(A),field2(A),P25) = bot_bot(set(A)) )
           => ( aa(set(product_prod(B,B)),set(B),field2(B),R23) = bot_bot(set(B)) ) )
         => pp(aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),bool,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P25)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ).

% cexp_mono2'
tff(fact_8135_ordLeq__cexp1,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
      ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),Q3),Q3)
       => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B))))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),Q3),bNF_Cardinal_cexp(B,A,Q3,R3))),bNF_Wellorder_ordLeq(B,fun(A,B)))) ) ) ).

% ordLeq_cexp1
tff(fact_8136_cexp__cong2,axiom,
    ! [B: $tType,C: $tType,A: $tType,P25: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P25),R23)),bNF_Wellorder_ordIso(A,B)))
     => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
       => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P25),P25)
         => pp(aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),bool,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P25)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordIso(fun(A,C),fun(B,C)))) ) ) ) ).

% cexp_cong2
tff(fact_8137_cexp__cong1,axiom,
    ! [B: $tType,A: $tType,C: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordIso(A,B)))
     => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
       => pp(aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),bool,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P13,Q3)),bNF_Cardinal_cexp(B,C,R12,Q3))),bNF_Wellorder_ordIso(fun(C,A),fun(C,B)))) ) ) ).

% cexp_cong1
tff(fact_8138_cexp__cong,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set(product_prod(A,A)),R12: set(product_prod(B,B)),P25: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R12)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P25),R23)),bNF_Wellorder_ordIso(C,D)))
       => ( bNF_Ca8970107618336181345der_on(D,aa(set(product_prod(D,D)),set(D),field2(D),R23),R23)
         => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),P25),P25)
           => pp(aa(set(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),bool,member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P13,P25)),bNF_Cardinal_cexp(B,D,R12,R23))),bNF_Wellorder_ordIso(fun(C,A),fun(D,B)))) ) ) ) ) ).

% cexp_cong
tff(fact_8139_cexp__mono2__Cnotzero,axiom,
    ! [B: $tType,C: $tType,A: $tType,P25: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P25),R23)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
       => ( ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P25),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),P25),P25) )
         => pp(aa(set(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),bool,member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P25)),bNF_Cardinal_cexp(C,B,Q3,R23))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ).

% cexp_mono2_Cnotzero
tff(fact_8140_ordLeq__cexp2,axiom,
    ! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3)),bNF_Wellorder_ordLeq(bool,A)))
     => ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3)
       => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A))))),bool,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)))),R3),bNF_Cardinal_cexp(A,B,Q3,R3))),bNF_Wellorder_ordLeq(B,fun(B,A)))) ) ) ).

% ordLeq_cexp2
tff(fact_8141_Cfinite__cexp__Cinfinite,axiom,
    ! [A: $tType,B: $tType,S2: set(product_prod(A,A)),T6: 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,T6)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),T6),T6) )
       => pp(aa(set(product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool))))),bool,member(product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool)))),aa(set(product_prod(fun(B,bool),fun(B,bool))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool)))),aa(set(product_prod(fun(B,A),fun(B,A))),fun(set(product_prod(fun(B,bool),fun(B,bool))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool))))),product_Pair(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool)))),bNF_Cardinal_cexp(A,B,S2,T6)),bNF_Cardinal_cexp(bool,B,bNF_Cardinal_ctwo,T6))),bNF_Wellorder_ordLeq(fun(B,A),fun(B,bool)))) ) ) ).

% Cfinite_cexp_Cinfinite
tff(fact_8142_ctwo__Cnotzero,axiom,
    ( ~ pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(bool,bool)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(bool,bool))),aa(set(product_prod(bool,bool)),product_prod(set(product_prod(bool,bool)),set(product_prod(bool,bool))),aa(set(product_prod(bool,bool)),fun(set(product_prod(bool,bool)),product_prod(set(product_prod(bool,bool)),set(product_prod(bool,bool)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(bool,bool))),bNF_Cardinal_ctwo),bNF_Cardinal_czero(bool))),bNF_Wellorder_ordIso(bool,bool)))
    & bNF_Ca8970107618336181345der_on(bool,aa(set(product_prod(bool,bool)),set(bool),field2(bool),bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).

% ctwo_Cnotzero
tff(fact_8143_ordLess__ctwo__cexp,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(A,A)),fun(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool))))),product_Pair(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool)))),R3),bNF_Cardinal_cexp(bool,A,bNF_Cardinal_ctwo,R3))),bNF_We4044943003108391690rdLess(A,fun(A,bool)))) ) ).

% ordLess_ctwo_cexp
tff(fact_8144_ctwo__not__czero,axiom,
    ! [A: $tType] : ~ pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(bool,A))) ).

% ctwo_not_czero
tff(fact_8145_ctwo__ordLeq__Cinfinite,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R3)),bNF_Wellorder_ordLeq(bool,A))) ) ).

% ctwo_ordLeq_Cinfinite
tff(fact_8146_ctwo__ordLess__Cinfinite,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R3)),bNF_We4044943003108391690rdLess(bool,A))) ) ).

% ctwo_ordLess_Cinfinite
tff(fact_8147_Cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3)),bNF_Wellorder_ordLeq(bool,A)))
     => ( ( bNF_Ca4139267488887388095finite(B,R3)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3) )
       => ( bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R3))
          & 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,Q3,R3)),bNF_Cardinal_cexp(A,B,Q3,R3)) ) ) ) ).

% Cinfinite_cexp
tff(fact_8148_cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3)),bNF_Wellorder_ordLeq(bool,A)))
     => ( ( bNF_Ca4139267488887388095finite(B,R3)
          & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R3),R3) )
       => bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R3)) ) ) ).

% cinfinite_cexp
tff(fact_8149_card__of__Plus__Times__aux,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,A5: set(A),B4: set(B)] :
      ( ( ( A1 != A22 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A22),bot_bot(set(A))))),A5)) )
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,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,A5,B4))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).

% card_of_Plus_Times_aux
tff(fact_8150_natLeq__ordLeq__cinfinite,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3) )
     => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca8665028551170535155natLeq),R3)),bNF_Wellorder_ordLeq(nat,A))) ) ).

% natLeq_ordLeq_cinfinite
tff(fact_8151_ordIso__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),P3: set(product_prod(C,C)),P10: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P10)),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,aa(set(product_prod(A,A)),set(A),field2(A),R3),aa(set(product_prod(C,C)),set(C),field2(C),P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,aa(set(product_prod(B,B)),set(B),field2(B),R6),aa(set(product_prod(D,D)),set(D),field2(D),P10))))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordIso_Plus_cong
tff(fact_8152_ordLeq__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),P3: set(product_prod(C,C)),P10: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P10)),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,aa(set(product_prod(A,A)),set(A),field2(A),R3),aa(set(product_prod(C,C)),set(C),field2(C),P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,aa(set(product_prod(B,B)),set(B),field2(B),R6),aa(set(product_prod(D,D)),set(D),field2(D),P10))))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordLeq_Plus_mono
tff(fact_8153_ordIso__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,aa(set(product_prod(A,A)),set(A),field2(A),R3),C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,aa(set(product_prod(B,B)),set(B),field2(B),R6),C4)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordIso_Plus_cong1
tff(fact_8154_ordIso__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),A5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A5,aa(set(product_prod(A,A)),set(A),field2(A),R3)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A5,aa(set(product_prod(B,B)),set(B),field2(B),R6))))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordIso_Plus_cong2
tff(fact_8155_ordLeq__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,aa(set(product_prod(A,A)),set(A),field2(A),R3),C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,aa(set(product_prod(B,B)),set(B),field2(B),R6),C4)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordLeq_Plus_mono1
tff(fact_8156_ordLeq__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),A5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A5,aa(set(product_prod(A,A)),set(A),field2(A),R3)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A5,aa(set(product_prod(B,B)),set(B),field2(B),R6))))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordLeq_Plus_mono2
tff(fact_8157_Card__order__Plus2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A5: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),R3),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A5,aa(set(product_prod(A,A)),set(A),field2(A),R3))))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ) ).

% Card_order_Plus2
tff(fact_8158_Card__order__Plus1,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),B4: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),R3),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R3),B4)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ) ).

% Card_order_Plus1
tff(fact_8159_ctwo__ordLess__natLeq,axiom,
    pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(bool,bool)),set(product_prod(nat,nat))),aa(set(product_prod(bool,bool)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(bool,bool)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(nat,nat))),bNF_Cardinal_ctwo),bNF_Ca8665028551170535155natLeq)),bNF_We4044943003108391690rdLess(bool,nat))) ).

% ctwo_ordLess_natLeq
tff(fact_8160_card__of__Plus1,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A5,B4)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ).

% card_of_Plus1
tff(fact_8161_card__of__Plus2,axiom,
    ! [B: $tType,A: $tType,B4: set(A),A5: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,B4)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A5,B4)))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ).

% card_of_Plus2
tff(fact_8162_card__of__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C),D4: set(D)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C4)),bNF_Ca6860139660246222851ard_of(D,D4))),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A5,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B4,D4)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_cong
tff(fact_8163_card__of__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C),D4: set(D)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C4)),bNF_Ca6860139660246222851ard_of(D,D4))),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A5,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B4,D4)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_mono
tff(fact_8164_card__of__Plus__assoc,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),bool,member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),fun(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),bNF_Ca6860139660246222851ard_of(sum_sum(sum_sum(A,B),C),sum_Plus(sum_sum(A,B),C,sum_Plus(A,B,A5,B4),C4))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,C)),sum_Plus(A,sum_sum(B,C),A5,sum_Plus(B,C,B4,C4))))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C))))) ).

% card_of_Plus_assoc
tff(fact_8165_card__of__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A5,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B4,C4)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_cong1
tff(fact_8166_card__of__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C4,A5))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C4,B4)))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_cong2
tff(fact_8167_card__of__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A5,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B4,C4)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_mono1
tff(fact_8168_card__of__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(B,B4))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C4,A5))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C4,B4)))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_mono2
tff(fact_8169_card__of__Plus__commute,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] : pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,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,A5,B4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B4,A5)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A)))) ).

% card_of_Plus_commute
tff(fact_8170_card__of__Plus__Times__bool,axiom,
    ! [A: $tType,A5: set(A)] : pp(aa(set(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),bool,member(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),aa(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),fun(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),product_Pair(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A5,A5))),bNF_Ca6860139660246222851ard_of(product_prod(A,bool),product_Sigma(A,bool,A5,aTP_Lamp_art(A,set(bool)))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,bool)))) ).

% card_of_Plus_Times_bool
tff(fact_8171_card__of__Times__Plus__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),bool,member(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),fun(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),product_Pair(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),bNF_Ca6860139660246222851ard_of(product_prod(A,sum_sum(B,C)),product_Sigma(A,sum_sum(B,C),A5,aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_aru(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),B4),C4)))),bNF_Ca6860139660246222851ard_of(sum_sum(product_prod(A,B),product_prod(A,C)),sum_Plus(product_prod(A,B),product_prod(A,C),product_Sigma(A,B,A5,aTP_Lamp_xt(set(B),fun(A,set(B)),B4)),product_Sigma(A,C,A5,aTP_Lamp_yh(set(C),fun(A,set(C)),C4)))))),bNF_Wellorder_ordIso(product_prod(A,sum_sum(B,C)),sum_sum(product_prod(A,B),product_prod(A,C))))) ).

% card_of_Times_Plus_distrib
tff(fact_8172_card__of__Plus__empty2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A5)))),bNF_Wellorder_ordIso(A,sum_sum(B,A)))) ).

% card_of_Plus_empty2
tff(fact_8173_card__of__Plus__empty1,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A5,bot_bot(set(B)))))),bNF_Wellorder_ordIso(A,sum_sum(A,B)))) ).

% card_of_Plus_empty1
tff(fact_8174_card__of__Un__Plus__ordLeq,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A5,B4)))),bNF_Wellorder_ordLeq(A,sum_sum(A,A)))) ).

% card_of_Un_Plus_ordLeq
tff(fact_8175_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less_eq(nat))) ).

% natLeq_def
tff(fact_8176_natLeq__underS__less,axiom,
    ! [N2: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N2) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2)) ).

% natLeq_underS_less
tff(fact_8177_card__of__Plus__infinite2,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,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,B4,A5))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ).

% card_of_Plus_infinite2
tff(fact_8178_card__of__Plus__infinite1,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,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,A5,B4))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ).

% card_of_Plus_infinite1
tff(fact_8179_card__of__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B4)),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,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,A5,B4))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
          & pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,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,B4,A5))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ).

% card_of_Plus_infinite
tff(fact_8180_card__of__Plus__ordLess__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,C4: set(A),A5: set(B),B4: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),C4))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A5)),bNF_Ca6860139660246222851ard_of(A,C4))),bNF_We4044943003108391690rdLess(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B4)),bNF_Ca6860139660246222851ard_of(A,C4))),bNF_We4044943003108391690rdLess(C,A)))
         => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A5,B4))),bNF_Ca6860139660246222851ard_of(A,C4))),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ).

% card_of_Plus_ordLess_infinite
tff(fact_8181_finite__iff__ordLess__natLeq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(A,A)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(A,A)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(A,A5)),bNF_Ca8665028551170535155natLeq)),bNF_We4044943003108391690rdLess(A,nat))) ) ).

% finite_iff_ordLess_natLeq
tff(fact_8182_Restr__natLeq,axiom,
    ! [N2: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),N2)),aTP_Lamp_arv(nat,fun(nat,set(nat)),N2))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ajh(nat,fun(nat,fun(nat,bool)),N2))) ).

% Restr_natLeq
tff(fact_8183_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_agn(product_prod(int,int),product_prod(int,int)),Uu) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),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_8184_ATP_Olambda__2,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_nl(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_2
tff(fact_8185_ATP_Olambda__3,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aa(nat,bool),Uu))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),lim(product_unit,h2)))
        & ~ pp(aa(set(nat),bool,member(nat,Uu),as1)) ) ) ).

% ATP.lambda_3
tff(fact_8186_ATP_Olambda__4,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ab(nat,bool),Uu))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),lim(product_unit,h2)))
        & ~ pp(aa(set(nat),bool,member(nat,Uu),as)) ) ) ).

% ATP.lambda_4
tff(fact_8187_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_uh(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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_5
tff(fact_8188_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_agr(product_prod(int,int),A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_6
tff(fact_8189_ATP_Olambda__7,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_agp(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_7
tff(fact_8190_ATP_Olambda__8,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_agw(product_prod(A,A),bool),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_8
tff(fact_8191_ATP_Olambda__9,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_ajx(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_9
tff(fact_8192_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( pp(aa(nat,bool,aTP_Lamp_rz(nat,bool),Uu))
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_10
tff(fact_8193_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_pu(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_11
tff(fact_8194_ATP_Olambda__12,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_agt(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).

% ATP.lambda_12
tff(fact_8195_ATP_Olambda__13,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_wu(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_13
tff(fact_8196_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_uv(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_14
tff(fact_8197_ATP_Olambda__15,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_cd(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_15
tff(fact_8198_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_ahj(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_16
tff(fact_8199_ATP_Olambda__17,axiom,
    ! [A: $tType,Uu: A] : aa(A,multiset(A),aTP_Lamp_akb(A,multiset(A)),Uu) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),zero_zero(multiset(A))) ).

% ATP.lambda_17
tff(fact_8200_ATP_Olambda__18,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,list(A),aTP_Lamp_ux(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).

% ATP.lambda_18
tff(fact_8201_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_te(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_19
tff(fact_8202_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_pz(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_20
tff(fact_8203_ATP_Olambda__21,axiom,
    ! [B: $tType,D: $tType,Uu: fun(D,B)] : aa(fun(D,B),set(B),aTP_Lamp_amt(fun(D,B),set(B)),Uu) = aa(set(D),set(B),image2(D,B,Uu),top_top(set(D))) ).

% ATP.lambda_21
tff(fact_8204_ATP_Olambda__22,axiom,
    ! [A: $tType,D: $tType,Uu: fun(D,A)] : aa(fun(D,A),set(A),aTP_Lamp_ams(fun(D,A),set(A)),Uu) = aa(set(D),set(A),image2(D,A,Uu),top_top(set(D))) ).

% ATP.lambda_22
tff(fact_8205_ATP_Olambda__23,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_akq(list(B),bool),Uu))
    <=> ( Uu = nil(B) ) ) ).

% ATP.lambda_23
tff(fact_8206_ATP_Olambda__24,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_akp(list(A),bool),Uu))
    <=> ( Uu = nil(A) ) ) ).

% ATP.lambda_24
tff(fact_8207_ATP_Olambda__25,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_agl(product_prod(int,int),bool),Uu))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))) ) ).

% ATP.lambda_25
tff(fact_8208_ATP_Olambda__26,axiom,
    ! [Uu: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_apz(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_apy(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu)) ).

% ATP.lambda_26
tff(fact_8209_ATP_Olambda__27,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_adr(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_27
tff(fact_8210_ATP_Olambda__28,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_ug(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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_28
tff(fact_8211_ATP_Olambda__29,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_ads(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_29
tff(fact_8212_ATP_Olambda__30,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_si(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_30
tff(fact_8213_ATP_Olambda__31,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_abp(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = aa(fun(A,fun(B,fun(A,option(B)))),fun(product_prod(A,B),fun(A,option(B))),product_case_prod(A,B,fun(A,option(B))),aTP_Lamp_abo(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_31
tff(fact_8214_ATP_Olambda__32,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_abl(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_abk(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_32
tff(fact_8215_ATP_Olambda__33,axiom,
    ! [A: $tType,Uu: multiset(A)] : aa(multiset(A),set(A),aTP_Lamp_alt(multiset(A),set(A)),Uu) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_all(multiset(A),fun(A,bool),Uu)) ).

% ATP.lambda_33
tff(fact_8216_ATP_Olambda__34,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_ry(nat,set(nat)),Uu) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_il(nat,fun(nat,bool),Uu)) ).

% ATP.lambda_34
tff(fact_8217_ATP_Olambda__35,axiom,
    ! [B: $tType,Uu: fun(B,nat)] :
      ( pp(aa(fun(B,nat),bool,aTP_Lamp_anx(fun(B,nat),bool),Uu))
    <=> pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_anw(fun(B,nat),fun(B,bool),Uu)))) ) ).

% ATP.lambda_35
tff(fact_8218_ATP_Olambda__36,axiom,
    ! [A: $tType,Uu: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aTP_Lamp_alo(fun(A,nat),bool),Uu))
    <=> pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_gi(fun(A,nat),fun(A,bool),Uu)))) ) ).

% ATP.lambda_36
tff(fact_8219_ATP_Olambda__37,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_aev(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_aeu(fun(A,B),fun(A,fun(B,bool)),Uu))) ).

% ATP.lambda_37
tff(fact_8220_ATP_Olambda__38,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_adq(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_38
tff(fact_8221_ATP_Olambda__39,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_adn(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_39
tff(fact_8222_ATP_Olambda__40,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_pp(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_40
tff(fact_8223_ATP_Olambda__41,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_jd(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_41
tff(fact_8224_ATP_Olambda__42,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_afs(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_42
tff(fact_8225_ATP_Olambda__43,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agb(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_43
tff(fact_8226_ATP_Olambda__44,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_iy(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_44
tff(fact_8227_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_akw(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_45
tff(fact_8228_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_akv(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_46
tff(fact_8229_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ady(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_47
tff(fact_8230_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_adx(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_48
tff(fact_8231_ATP_Olambda__49,axiom,
    ! [Uu: int] : aa(int,nat,aTP_Lamp_sa(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_49
tff(fact_8232_ATP_Olambda__50,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_agv(product_prod(A,A),bool),Uu))
    <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% ATP.lambda_50
tff(fact_8233_ATP_Olambda__51,axiom,
    ! [A: $tType,B: $tType,Uu: product_prod(product_prod(bool,A),product_prod(bool,B))] :
      ( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_aar(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu))
    <=> ? [X4: A,Y5: B] : Uu = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fTrue),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y5)) ) ).

% ATP.lambda_51
tff(fact_8234_ATP_Olambda__52,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_kp(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_kn(nat,fun(num,option(num)),Uua),aTP_Lamp_ko(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_52
tff(fact_8235_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ij(nat,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_53
tff(fact_8236_ATP_Olambda__54,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] : aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_wy(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_54
tff(fact_8237_ATP_Olambda__55,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,option(A),aTP_Lamp_lw(list(A),fun(nat,option(A)),Uu),Uua) = if(option(A),aa(nat,bool,aa(nat,fun(nat,bool),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_55
tff(fact_8238_ATP_Olambda__56,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_lk(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(code_integer,int,code_int_of_integer,Uu)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(code_integer,int,code_int_of_integer,Uu))),one_one(int))) ).

% ATP.lambda_56
tff(fact_8239_ATP_Olambda__57,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_lj(code_integer,fun(code_integer,num)),Uu),Uua) = if(num,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu))),one2)) ).

% ATP.lambda_57
tff(fact_8240_ATP_Olambda__58,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_lp(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one(nat))) ).

% ATP.lambda_58
tff(fact_8241_ATP_Olambda__59,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agg(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_59
tff(fact_8242_ATP_Olambda__60,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ago(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),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_60
tff(fact_8243_ATP_Olambda__61,axiom,
    ! [A: $tType,Uu: set(fun(A,nat)),Uua: A] : aa(A,nat,aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_alz(set(fun(A,nat)),fun(A,nat)),Uu),Uua) = if(nat,aa(set(fun(A,nat)),bool,aa(set(fun(A,nat)),fun(set(fun(A,nat)),bool),fequal(set(fun(A,nat))),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_alp(A,fun(fun(A,nat),nat),Uua)),Uu))) ).

% ATP.lambda_61
tff(fact_8244_ATP_Olambda__62,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ii(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_62
tff(fact_8245_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajy(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ).

% ATP.lambda_63
tff(fact_8246_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_sf(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_64
tff(fact_8247_ATP_Olambda__65,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aku(list(list(A)),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(list(list(A)),bool,aa(list(A),fun(list(list(A)),bool),list_all2(A,list(A),aTP_Lamp_akt(A,fun(list(A),bool))),Uua),Uu)) ) ).

% ATP.lambda_65
tff(fact_8248_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_bg(option(A),fun(A,option(A)),Uu),Uua) = case_option(option(A),A,none(A),aTP_Lamp_bf(A,fun(A,option(A)),Uua),Uu) ) ).

% ATP.lambda_66
tff(fact_8249_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_bp(option(A),fun(A,bool),Uu),Uua))
        <=> pp(case_option(bool,A,fTrue,aa(A,fun(A,bool),aTP_Lamp_bo(A,fun(A,bool)),Uua),Uu)) ) ) ).

% ATP.lambda_67
tff(fact_8250_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_kq(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_kp(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_68
tff(fact_8251_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_bn(option(A),fun(A,bool),Uu),Uua))
        <=> pp(case_option(bool,A,fFalse,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_69
tff(fact_8252_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_kn(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_jd(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_70
tff(fact_8253_ATP_Olambda__71,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_aff(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua))) ) ).

% ATP.lambda_71
tff(fact_8254_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ada(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_72
tff(fact_8255_ATP_Olambda__73,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_vs(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vr(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_73
tff(fact_8256_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jj(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_ji(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_74
tff(fact_8257_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jh(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_jg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_75
tff(fact_8258_ATP_Olambda__76,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_agm(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_76
tff(fact_8259_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fi(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_77
tff(fact_8260_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hq(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_78
tff(fact_8261_ATP_Olambda__79,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_agq(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_79
tff(fact_8262_ATP_Olambda__80,axiom,
    ! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
      ( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_ara(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Uu),Uu)
        & order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),Uua),Uua)
        & ? [X_12: fun(A,A2)] : bNF_Wellorder_embedS(A,A2,Uu,Uua,X_12) ) ) ).

% ATP.lambda_80
tff(fact_8263_ATP_Olambda__81,axiom,
    ! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
      ( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_are(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Uu),Uu)
        & order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),Uua),Uua)
        & ? [X_12: fun(A,A2)] : bNF_Wellorder_embed(A,A2,Uu,Uua,X_12) ) ) ).

% ATP.lambda_81
tff(fact_8264_ATP_Olambda__82,axiom,
    ! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
      ( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_arf(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Uu),Uu)
        & order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),Uua),Uua)
        & ? [X_12: fun(A,A2)] : bNF_Wellorder_iso(A,A2,Uu,Uua,X_12) ) ) ).

% ATP.lambda_82
tff(fact_8265_ATP_Olambda__83,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_afi(rat,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu))
        & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_83
tff(fact_8266_ATP_Olambda__84,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_lt(set(A),fun(list(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu))
        & distinct(A,Uua) ) ) ).

% ATP.lambda_84
tff(fact_8267_ATP_Olambda__85,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_ls(set(A),fun(list(A),bool),Uu),Uua))
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
        & distinct(A,Uua) ) ) ).

% ATP.lambda_85
tff(fact_8268_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fh(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_86
tff(fact_8269_ATP_Olambda__87,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_agu(rat,fun(product_prod(int,int),bool),Uu),Uua))
    <=> ( ( Uu = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Uua)))
        & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).

% ATP.lambda_87
tff(fact_8270_ATP_Olambda__88,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ahb(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))),insert(set(A)),aa(set(A),set(A),image(A,A,Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% ATP.lambda_88
tff(fact_8271_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fb(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_89
tff(fact_8272_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hv(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_90
tff(fact_8273_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fe(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_91
tff(fact_8274_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_im(nat,fun(nat,bool)),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_92
tff(fact_8275_ATP_Olambda__93,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_nm(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_93
tff(fact_8276_ATP_Olambda__94,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_pv(set(option(A)),fun(option(A),bool),Uu),Uua))
    <=> ( pp(aa(set(option(A)),bool,member(option(A),Uua),Uu))
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_94
tff(fact_8277_ATP_Olambda__95,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fr(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% ATP.lambda_95
tff(fact_8278_ATP_Olambda__96,axiom,
    ! [Uu: set(int),Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_aht(set(int),fun(int,bool),Uu),Uua))
    <=> ( pp(aa(set(int),bool,member(int,Uua),Uu))
        & ! [X4: int] :
            ( pp(aa(set(int),bool,member(int,X4),Uu))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X4),Uua)) ) ) ) ).

% ATP.lambda_96
tff(fact_8279_ATP_Olambda__97,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ahu(set(set(A)),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,member(set(A),Uua),Uu))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
           => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),X4)) ) ) ) ).

% ATP.lambda_97
tff(fact_8280_ATP_Olambda__98,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_anr(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),Uu),Uua))
        & ! [A8: A,B13: A,C5: A] :
            ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A8),B13)),Uua))
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),C5)),Uu)) )
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A8),B13)),Uu)) ) ) ) ).

% ATP.lambda_98
tff(fact_8281_ATP_Olambda__99,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_aju(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & chain_subset(A,Uua) ) ) ).

% ATP.lambda_99
tff(fact_8282_ATP_Olambda__100,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_pl(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu))
        & pp(aa(set(A),bool,finite_finite2(A),Uua)) ) ) ).

% ATP.lambda_100
tff(fact_8283_ATP_Olambda__101,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_jq(set(A),fun(set(A),bool)),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uu),Uua))
        & pp(aa(set(A),bool,finite_finite2(A),Uua)) ) ) ).

% ATP.lambda_101
tff(fact_8284_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sh(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ).

% ATP.lambda_102
tff(fact_8285_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).

% ATP.lambda_103
tff(fact_8286_ATP_Olambda__104,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gt(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).

% ATP.lambda_104
tff(fact_8287_ATP_Olambda__105,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_sj(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))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_105
tff(fact_8288_ATP_Olambda__106,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_yv(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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_106
tff(fact_8289_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_hd(A,fun(A,bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,member(A,Uua),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).

% ATP.lambda_107
tff(fact_8290_ATP_Olambda__108,axiom,
    ! [D: $tType,B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(D,product_prod(A,B))] :
      ( pp(aa(fun(D,product_prod(A,B)),bool,aTP_Lamp_ahf(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(D),set(product_prod(A,B)),image2(D,product_prod(A,B),Uua),top_top(set(D)))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_108
tff(fact_8291_ATP_Olambda__109,axiom,
    ! [A: $tType,Uu: fun(set(A),bool),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_zu(fun(set(A),bool),fun(set(A),bool)),Uu),Uua))
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A13: set(A),A8: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A8),A13) )
            & pp(aa(set(A),bool,Uu,A13)) ) ) ) ).

% ATP.lambda_109
tff(fact_8292_ATP_Olambda__110,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aTP_Lamp_arl(set(B),fun(fun(A,B),bool),Uu),Uua))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,Uua),top_top(set(A)))),Uu)) ) ).

% ATP.lambda_110
tff(fact_8293_ATP_Olambda__111,axiom,
    ! [D: $tType,A: $tType,Uu: set(A),Uua: fun(D,A)] :
      ( pp(aa(fun(D,A),bool,aTP_Lamp_aqy(set(A),fun(fun(D,A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,Uua),top_top(set(D)))),Uu)) ) ).

% ATP.lambda_111
tff(fact_8294_ATP_Olambda__112,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_aha(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)),insert(A),Uua),bot_bot(set(A))),Uu) ).

% ATP.lambda_112
tff(fact_8295_ATP_Olambda__113,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_xo(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).

% ATP.lambda_113
tff(fact_8296_ATP_Olambda__114,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_zb(nat,fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).

% ATP.lambda_114
tff(fact_8297_ATP_Olambda__115,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_xp(fun(C,product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(C),set(B),image2(C,B,aa(fun(C,product_prod(A,B)),fun(C,B),comp(product_prod(A,B),B,C,product_snd(A,B)),Uu)),top_top(set(C))) ).

% ATP.lambda_115
tff(fact_8298_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_116
tff(fact_8299_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_117
tff(fact_8300_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_118
tff(fact_8301_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hn(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_119
tff(fact_8302_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hj(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_120
tff(fact_8303_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fp(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_121
tff(fact_8304_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dk(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_122
tff(fact_8305_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dj(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_123
tff(fact_8306_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ey(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_124
tff(fact_8307_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_akn(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y5: A] :
                ( pp(aa(A,bool,Uu,Y5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5)) ) ) ) ) ).

% ATP.lambda_125
tff(fact_8308_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ajb(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y5: A] :
                ( pp(aa(A,bool,Uu,Y5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),Uua)) ) ) ) ) ).

% ATP.lambda_126
tff(fact_8309_ATP_Olambda__127,axiom,
    ! [A: $tType,Uu: fun(multiset(A),bool),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_ajd(fun(multiset(A),bool),fun(multiset(A),bool),Uu),Uua))
    <=> ( pp(aa(multiset(A),bool,Uu,Uua))
        & ! [Y5: multiset(A)] :
            ( pp(aa(multiset(A),bool,Uu,Y5))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Uua),Y5)) ) ) ) ).

% ATP.lambda_127
tff(fact_8310_ATP_Olambda__128,axiom,
    ! [A: $tType,Uu: fun(multiset(A),bool),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_ajo(fun(multiset(A),bool),fun(multiset(A),bool),Uu),Uua))
    <=> ( pp(aa(multiset(A),bool,Uu,Uua))
        & ! [Y5: multiset(A)] :
            ( pp(aa(multiset(A),bool,Uu,Y5))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Y5),Uua)) ) ) ) ).

% ATP.lambda_128
tff(fact_8311_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kg(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_129
tff(fact_8312_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kf(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_130
tff(fact_8313_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_add(fun(A,A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua)) ) ) ).

% ATP.lambda_131
tff(fact_8314_ATP_Olambda__132,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_ajz(fun(A,nat),fun(A,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(A,nat,Uu,Uua)),Uua) ).

% ATP.lambda_132
tff(fact_8315_ATP_Olambda__133,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_aer(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_133
tff(fact_8316_ATP_Olambda__134,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_um(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_134
tff(fact_8317_ATP_Olambda__135,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_qi(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_135
tff(fact_8318_ATP_Olambda__136,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_gy(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_136
tff(fact_8319_ATP_Olambda__137,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_hc(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_137
tff(fact_8320_ATP_Olambda__138,axiom,
    ! [A: $tType,Uu: list(fun(A,nat)),Uua: A] : aa(A,list(nat),aTP_Lamp_aob(list(fun(A,nat)),fun(A,list(nat)),Uu),Uua) = aa(list(fun(A,nat)),list(nat),map(fun(A,nat),nat,aTP_Lamp_alp(A,fun(fun(A,nat),nat),Uua)),Uu) ).

% ATP.lambda_138
tff(fact_8321_ATP_Olambda__139,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_ur(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_uq(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_139
tff(fact_8322_ATP_Olambda__140,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_jb(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_140
tff(fact_8323_ATP_Olambda__141,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aeb(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).

% ATP.lambda_141
tff(fact_8324_ATP_Olambda__142,axiom,
    ! [Uu: assn,Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_ch(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,Uua))
        & ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uu),Uua)) ) ) ).

% ATP.lambda_142
tff(fact_8325_ATP_Olambda__143,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: multiset(product_prod(A,B))] :
      ( pp(aa(multiset(product_prod(A,B)),bool,aTP_Lamp_alj(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_143
tff(fact_8326_ATP_Olambda__144,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: list(product_prod(A,B))] :
      ( pp(aa(list(product_prod(A,B)),bool,aTP_Lamp_ako(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_144
tff(fact_8327_ATP_Olambda__145,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(set(A))] : aa(set(set(A)),set(A),aTP_Lamp_aoc(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uua)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))) ).

% ATP.lambda_145
tff(fact_8328_ATP_Olambda__146,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_aoh(fun(A,fun(B,bool)),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),Uua)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))) ).

% ATP.lambda_146
tff(fact_8329_ATP_Olambda__147,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
          ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_wj(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_147
tff(fact_8330_ATP_Olambda__148,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
          ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_wl(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_148
tff(fact_8331_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_rc(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_149
tff(fact_8332_ATP_Olambda__150,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_ahh(num,fun(num,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Uu)),aa(num,int,numeral_numeral(int),Uua)) ).

% ATP.lambda_150
tff(fact_8333_ATP_Olambda__151,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_vn(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua)),product_lists(A,Uu)) ).

% ATP.lambda_151
tff(fact_8334_ATP_Olambda__152,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ahr(list(A),fun(list(A),bool)),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_152
tff(fact_8335_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_acd(list(A),fun(list(A),bool),Uu),Uua))
    <=> ( mset(A,Uua) = mset(A,Uu) ) ) ).

% ATP.lambda_153
tff(fact_8336_ATP_Olambda__154,axiom,
    ! [A: $tType,Uu: set(A),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_aqz(set(A),fun(multiset(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(multiset(A),set(A),set_mset(A),Uua)),Uu)) ) ).

% ATP.lambda_154
tff(fact_8337_ATP_Olambda__155,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_ajf(set(A),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)) ) ).

% ATP.lambda_155
tff(fact_8338_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dl(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_156
tff(fact_8339_ATP_Olambda__157,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_up(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).

% ATP.lambda_157
tff(fact_8340_ATP_Olambda__158,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_acb(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)),Uu)) ) ).

% ATP.lambda_158
tff(fact_8341_ATP_Olambda__159,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_acj(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,aa(nat,nat,suc,Uua)),Uu)) ) ).

% ATP.lambda_159
tff(fact_8342_ATP_Olambda__160,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aji(nat,fun(list(A),bool),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).

% ATP.lambda_160
tff(fact_8343_ATP_Olambda__161,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_jt(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_161
tff(fact_8344_ATP_Olambda__162,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_jc(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_162
tff(fact_8345_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fj(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_163
tff(fact_8346_ATP_Olambda__164,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_sy(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)),insert(A),Uu)),Uua)) ).

% ATP.lambda_164
tff(fact_8347_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ti(list(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% ATP.lambda_165
tff(fact_8348_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_tj(list(A),fun(A,bool),Uu),Uua))
        <=> ( Uua = aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% ATP.lambda_166
tff(fact_8349_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ga(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_167
tff(fact_8350_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: multiset(A),Uua: A] : aa(A,A,aTP_Lamp_apv(multiset(A),fun(A,A),Uu),Uua) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uua)) ) ).

% ATP.lambda_168
tff(fact_8351_ATP_Olambda__169,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_pw(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_169
tff(fact_8352_ATP_Olambda__170,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_ql(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_170
tff(fact_8353_ATP_Olambda__171,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_ahd(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),image(B,A,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_171
tff(fact_8354_ATP_Olambda__172,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_acx(fun(A,B),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_172
tff(fact_8355_ATP_Olambda__173,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_afg(set(A),fun(A,bool),Uu),Uua))
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_173
tff(fact_8356_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aqo(fun(A,A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(A,A,Uu,Uua))) ) ) ).

% ATP.lambda_174
tff(fact_8357_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_vw(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_175
tff(fact_8358_ATP_Olambda__176,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_wh(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_176
tff(fact_8359_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_mf(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_177
tff(fact_8360_ATP_Olambda__178,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_afo(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_178
tff(fact_8361_ATP_Olambda__179,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_je(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).

% ATP.lambda_179
tff(fact_8362_ATP_Olambda__180,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_yu(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ).

% ATP.lambda_180
tff(fact_8363_ATP_Olambda__181,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_vm(nat,fun(list(A),bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_181
tff(fact_8364_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ee(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_182
tff(fact_8365_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hz(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_183
tff(fact_8366_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fc(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_184
tff(fact_8367_ATP_Olambda__185,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_adf(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_185
tff(fact_8368_ATP_Olambda__186,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_akt(A,fun(list(A),bool)),Uu),Uua))
    <=> pp(aa(set(A),bool,member(A,Uu),aa(list(A),set(A),set2(A),Uua))) ) ).

% ATP.lambda_186
tff(fact_8369_ATP_Olambda__187,axiom,
    ! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_rw(list(set(A)),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(set(A)),bool,member(set(A),Uua),aa(list(set(A)),set(set(A)),set2(set(A)),Uu))) ) ).

% ATP.lambda_187
tff(fact_8370_ATP_Olambda__188,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_sr(list(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,member(A,Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_188
tff(fact_8371_ATP_Olambda__189,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_ajj(nat,fun(list(A),bool),Uu),Uua))
    <=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_189
tff(fact_8372_ATP_Olambda__190,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_wd(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_190
tff(fact_8373_ATP_Olambda__191,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_wc(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = last(A,Uu) ) ) ).

% ATP.lambda_191
tff(fact_8374_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_rr(nat,fun(nat,bool)),Uu),Uua))
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_192
tff(fact_8375_ATP_Olambda__193,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ie(set(A),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ).

% ATP.lambda_193
tff(fact_8376_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ca(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_194
tff(fact_8377_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_al(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_195
tff(fact_8378_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ax(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_196
tff(fact_8379_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_av(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_197
tff(fact_8380_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_se(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_198
tff(fact_8381_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_es(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_199
tff(fact_8382_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pm(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_200
tff(fact_8383_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_201
tff(fact_8384_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aen(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_202
tff(fact_8385_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_203
tff(fact_8386_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_am(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_204
tff(fact_8387_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aoq(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_205
tff(fact_8388_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ay(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_206
tff(fact_8389_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aw(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_207
tff(fact_8390_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bo(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_208
tff(fact_8391_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_adt(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_209
tff(fact_8392_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_jz(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_210
tff(fact_8393_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_211
tff(fact_8394_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qm(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_212
tff(fact_8395_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cs(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_213
tff(fact_8396_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_214
tff(fact_8397_ATP_Olambda__215,axiom,
    ! [A: $tType,Uu: A,Uua: multiset(A)] : aa(multiset(A),nat,aTP_Lamp_alq(A,fun(multiset(A),nat),Uu),Uua) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uu) ).

% ATP.lambda_215
tff(fact_8398_ATP_Olambda__216,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_acu(multiset(A),fun(multiset(A),bool)),Uu),Uua))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Uua),Uu)) ) ).

% ATP.lambda_216
tff(fact_8399_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_217
tff(fact_8400_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_mv(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).

% ATP.lambda_218
tff(fact_8401_ATP_Olambda__219,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_no(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_219
tff(fact_8402_ATP_Olambda__220,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_kl(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_220
tff(fact_8403_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_vt(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_221
tff(fact_8404_ATP_Olambda__222,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_if(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_222
tff(fact_8405_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aeq(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_223
tff(fact_8406_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cr(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_224
tff(fact_8407_ATP_Olambda__225,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_acv(multiset(A),fun(multiset(A),bool)),Uu),Uua))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Uua),Uu)) ) ).

% ATP.lambda_225
tff(fact_8408_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: ref(A)] : aa(ref(A),assn,aa(A,fun(ref(A),assn),aTP_Lamp_an(A,fun(ref(A),assn)),Uu),Uua) = sngr_assn(A,Uua,Uu) ) ).

% ATP.lambda_226
tff(fact_8409_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aa(list(A),fun(array(A),assn),aTP_Lamp_ao(list(A),fun(array(A),assn)),Uu),Uua) = snga_assn(A,Uua,Uu) ) ).

% ATP.lambda_227
tff(fact_8410_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_il(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).

% ATP.lambda_228
tff(fact_8411_ATP_Olambda__229,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_ix(int,fun(int,bool),Uu),Uua))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uua),Uu)) ) ).

% ATP.lambda_229
tff(fact_8412_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ik(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),Uu)) ) ) ).

% ATP.lambda_230
tff(fact_8413_ATP_Olambda__231,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_yi(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).

% ATP.lambda_231
tff(fact_8414_ATP_Olambda__232,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_rb(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_232
tff(fact_8415_ATP_Olambda__233,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_wk(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_233
tff(fact_8416_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_aco(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_234
tff(fact_8417_ATP_Olambda__235,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_wx(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_235
tff(fact_8418_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gb(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_236
tff(fact_8419_ATP_Olambda__237,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_uq(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_237
tff(fact_8420_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),A,aTP_Lamp_qy(ref(A),fun(heap_ext(product_unit),A),Uu),Uua) = ref_get(A,Uua,Uu) ) ).

% ATP.lambda_238
tff(fact_8421_ATP_Olambda__239,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_akh(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,Uua),Uu)) ) ).

% ATP.lambda_239
tff(fact_8422_ATP_Olambda__240,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_anv(set(B),fun(B,bool),Uu),Uua))
    <=> pp(aa(set(B),bool,member(B,Uua),Uu)) ) ).

% ATP.lambda_240
tff(fact_8423_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_akk(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ) ).

% ATP.lambda_241
tff(fact_8424_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_akm(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ) ).

% ATP.lambda_242
tff(fact_8425_ATP_Olambda__243,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),Uu),Uua))
    <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ).

% ATP.lambda_243
tff(fact_8426_ATP_Olambda__244,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_xn(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Uua),Uu) ).

% ATP.lambda_244
tff(fact_8427_ATP_Olambda__245,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_sz(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_245
tff(fact_8428_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_vl(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_246
tff(fact_8429_ATP_Olambda__247,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(C,set(B))] : aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_ael(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua) = aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,Uua),Uu) ).

% ATP.lambda_247
tff(fact_8430_ATP_Olambda__248,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_br(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_248
tff(fact_8431_ATP_Olambda__249,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ut(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_249
tff(fact_8432_ATP_Olambda__250,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_vk(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_250
tff(fact_8433_ATP_Olambda__251,axiom,
    ! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_aap(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_251
tff(fact_8434_ATP_Olambda__252,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_all(multiset(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uua))) ) ).

% ATP.lambda_252
tff(fact_8435_ATP_Olambda__253,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_anw(fun(B,nat),fun(B,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(B,nat,Uu,Uua))) ) ).

% ATP.lambda_253
tff(fact_8436_ATP_Olambda__254,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_gi(fun(A,nat),fun(A,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua))) ) ).

% ATP.lambda_254
tff(fact_8437_ATP_Olambda__255,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_alx(set(multiset(A)),fun(A,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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_alq(A,fun(multiset(A),nat),Uua)),Uu)))) ) ).

% ATP.lambda_255
tff(fact_8438_ATP_Olambda__256,axiom,
    ! [A: $tType,Uu: code_natural,Uua: A] :
      ( pp(aa(A,bool,aa(code_natural,fun(A,bool),aTP_Lamp_aqc(code_natural,fun(A,bool)),Uu),Uua))
    <=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),Uu)) ) ).

% ATP.lambda_256
tff(fact_8439_ATP_Olambda__257,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_yr(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_257
tff(fact_8440_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_abz(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).

% ATP.lambda_258
tff(fact_8441_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_aca(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_259
tff(fact_8442_ATP_Olambda__260,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_wr(fun(B,C),fun(product_prod(A,B),C),Uu),Uua) = aa(B,C,Uu,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).

% ATP.lambda_260
tff(fact_8443_ATP_Olambda__261,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_wm(fun(A,C),fun(product_prod(A,B),C),Uu),Uua) = aa(A,C,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_261
tff(fact_8444_ATP_Olambda__262,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aki(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_262
tff(fact_8445_ATP_Olambda__263,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_er(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_263
tff(fact_8446_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_264
tff(fact_8447_ATP_Olambda__265,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vy(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_265
tff(fact_8448_ATP_Olambda__266,axiom,
    ! [A: $tType,C: $tType,Uu: C,Uua: fun(C,set(set(A)))] : aa(fun(C,set(set(A))),set(set(A)),aTP_Lamp_arg(C,fun(fun(C,set(set(A))),set(set(A))),Uu),Uua) = aa(C,set(set(A)),Uua,Uu) ).

% ATP.lambda_266
tff(fact_8449_ATP_Olambda__267,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,set(A))] : aa(fun(B,set(A)),set(A),aTP_Lamp_aek(B,fun(fun(B,set(A)),set(A)),Uu),Uua) = aa(B,set(A),Uua,Uu) ).

% ATP.lambda_267
tff(fact_8450_ATP_Olambda__268,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_alp(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).

% ATP.lambda_268
tff(fact_8451_ATP_Olambda__269,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_ml(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_269
tff(fact_8452_ATP_Olambda__270,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_nr(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_270
tff(fact_8453_ATP_Olambda__271,axiom,
    ! [A: $tType,Uu: fun(product_unit,A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_app(fun(product_unit,A),fun(product_unit,A),Uu),Uua) = aa(product_unit,A,Uu,product_Unity) ).

% ATP.lambda_271
tff(fact_8454_ATP_Olambda__272,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aee(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_min(A),Uu),Uua)) ) ).

% ATP.lambda_272
tff(fact_8455_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_adv(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).

% ATP.lambda_273
tff(fact_8456_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aec(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_274
tff(fact_8457_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aed(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_275
tff(fact_8458_ATP_Olambda__276,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: option(multiset(A))] : aa(option(multiset(A)),option(multiset(A)),aa(multiset(A),fun(option(multiset(A)),option(multiset(A))),aTP_Lamp_amr(multiset(A),fun(option(multiset(A)),option(multiset(A)))),Uu),Uua) = aa(multiset(A),option(multiset(A)),some(multiset(A)),case_option(multiset(A),multiset(A),Uu,aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uu),Uua)) ).

% ATP.lambda_276
tff(fact_8459_ATP_Olambda__277,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: option(multiset(A))] : aa(option(multiset(A)),option(multiset(A)),aa(multiset(A),fun(option(multiset(A)),option(multiset(A))),aTP_Lamp_amj(multiset(A),fun(option(multiset(A)),option(multiset(A)))),Uu),Uua) = aa(multiset(A),option(multiset(A)),some(multiset(A)),case_option(multiset(A),multiset(A),Uu,aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uu),Uua)) ).

% ATP.lambda_277
tff(fact_8460_ATP_Olambda__278,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ko(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_278
tff(fact_8461_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_aif(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_aie(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_279
tff(fact_8462_ATP_Olambda__280,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C] : aa(C,fun(product_prod(B,A),C),aTP_Lamp_abt(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),Uu),Uua) = aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_abs(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).

% ATP.lambda_280
tff(fact_8463_ATP_Olambda__281,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(bool)),aTP_Lamp_uf(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Uu),Uua) = aa(fun(A,fun(list(A),option(bool))),fun(product_prod(A,list(A)),option(bool)),product_case_prod(A,list(A),option(bool)),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_ue(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua)) ).

% ATP.lambda_281
tff(fact_8464_ATP_Olambda__282,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_ud(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_uc(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_282
tff(fact_8465_ATP_Olambda__283,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),list(B)),aTP_Lamp_tn(fun(B,A),fun(list(B),fun(product_prod(list(B),list(B)),list(B))),Uu),Uua) = aa(fun(list(B),fun(list(B),list(B))),fun(product_prod(list(B),list(B)),list(B)),product_case_prod(list(B),list(B),list(B)),aa(list(B),fun(list(B),fun(list(B),list(B))),aTP_Lamp_tm(fun(B,A),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua)) ) ).

% ATP.lambda_283
tff(fact_8466_ATP_Olambda__284,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_rk(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_rj(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_284
tff(fact_8467_ATP_Olambda__285,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_ri(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_rh(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_285
tff(fact_8468_ATP_Olambda__286,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rg(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_rf(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_286
tff(fact_8469_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_re(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_rd(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_287
tff(fact_8470_ATP_Olambda__288,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_ra(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_qz(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_288
tff(fact_8471_ATP_Olambda__289,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_jo(fun(A,heap_Time_Heap(B)),fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua) = aa(fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(heap_ext(product_unit),nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_jn(fun(A,heap_Time_Heap(B)),fun(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).

% ATP.lambda_289
tff(fact_8472_ATP_Olambda__290,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_jl(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua) = aa(fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),product_case_prod(heap_ext(product_unit),nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_jk(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).

% ATP.lambda_290
tff(fact_8473_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,bool)] : aa(fun(A,bool),set(A),aTP_Lamp_aof(fun(A,fun(B,bool)),fun(fun(A,bool),set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aoe(fun(A,fun(B,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua)) ).

% ATP.lambda_291
tff(fact_8474_ATP_Olambda__292,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] : aa(B,set(A),aTP_Lamp_aah(fun(A,fun(B,bool)),fun(B,set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_aag(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua)) ).

% ATP.lambda_292
tff(fact_8475_ATP_Olambda__293,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_pg(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pf(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_293
tff(fact_8476_ATP_Olambda__294,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_ph(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_pd(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_294
tff(fact_8477_ATP_Olambda__295,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_pi(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pf(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_295
tff(fact_8478_ATP_Olambda__296,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_pe(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_pd(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_296
tff(fact_8479_ATP_Olambda__297,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_cc(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_297
tff(fact_8480_ATP_Olambda__298,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_wa(fun(A,option(B)),fun(A,bool),Uu),Uua))
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_298
tff(fact_8481_ATP_Olambda__299,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_yw(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_yv(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_299
tff(fact_8482_ATP_Olambda__300,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: A] : aa(A,nat,aTP_Lamp_ama(set(multiset(A)),fun(A,nat),Uu),Uua) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_alq(A,fun(multiset(A),nat),Uua)),Uu)) ).

% ATP.lambda_300
tff(fact_8483_ATP_Olambda__301,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_wz(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_wy(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).

% ATP.lambda_301
tff(fact_8484_ATP_Olambda__302,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_iq(nat,fun(nat,bool),Uu),Uua))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_302
tff(fact_8485_ATP_Olambda__303,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_mp(set(set(A)),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Sup_Sup(bool),aa(set(set(A)),set(bool),image2(set(A),bool,member(A,Uua)),Uu))) ) ).

% ATP.lambda_303
tff(fact_8486_ATP_Olambda__304,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_pk(set(set(A)),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Inf_Inf(bool),aa(set(set(A)),set(bool),image2(set(A),bool,member(A,Uua)),Uu))) ) ).

% ATP.lambda_304
tff(fact_8487_ATP_Olambda__305,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_apy(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_305
tff(fact_8488_ATP_Olambda__306,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_pn(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_306
tff(fact_8489_ATP_Olambda__307,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ck(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_307
tff(fact_8490_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ce(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_308
tff(fact_8491_ATP_Olambda__309,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_aqi(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(nat,A,nth(A,Uu),aa(code_natural,nat,code_nat_of_natural,Uua))) ).

% ATP.lambda_309
tff(fact_8492_ATP_Olambda__310,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_sq(list(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,member(A,Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_310
tff(fact_8493_ATP_Olambda__311,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_vx(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_311
tff(fact_8494_ATP_Olambda__312,axiom,
    ! [P7: $tType,O: $tType,Uu: fun(O,set(P7)),Uua: set(O)] : aa(set(O),set(P7),aTP_Lamp_mx(fun(O,set(P7)),fun(set(O),set(P7)),Uu),Uua) = aa(set(set(P7)),set(P7),complete_Sup_Sup(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),Uu),Uua)) ).

% ATP.lambda_312
tff(fact_8495_ATP_Olambda__313,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_aia(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,Uu),Uua)) ) ).

% ATP.lambda_313
tff(fact_8496_ATP_Olambda__314,axiom,
    ! [D: $tType,B: $tType,Uu: set(B),Uua: fun(B,set(D))] : aa(fun(B,set(D)),set(D),aa(set(B),fun(fun(B,set(D)),set(D)),aTP_Lamp_anc(set(B),fun(fun(B,set(D)),set(D))),Uu),Uua) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),Uua),Uu)) ).

% ATP.lambda_314
tff(fact_8497_ATP_Olambda__315,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_amz(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),Uu)) ) ).

% ATP.lambda_315
tff(fact_8498_ATP_Olambda__316,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(set(A),fun(fun(A,bool),bool),aTP_Lamp_aik(set(A),fun(fun(A,bool),bool)),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Sup_Sup(bool),aa(set(A),set(bool),image2(A,bool,Uua),Uu))) ) ).

% ATP.lambda_316
tff(fact_8499_ATP_Olambda__317,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: fun(A,set(C))] : aa(fun(A,set(C)),set(C),aa(set(A),fun(fun(A,set(C)),set(C)),aTP_Lamp_anb(set(A),fun(fun(A,set(C)),set(C))),Uu),Uua) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),Uua),Uu)) ).

% ATP.lambda_317
tff(fact_8500_ATP_Olambda__318,axiom,
    ! [C: $tType,A: $tType] :
      ( complete_Sup(C)
     => ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_amy(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,Uua),Uu)) ) ).

% ATP.lambda_318
tff(fact_8501_ATP_Olambda__319,axiom,
    ! [P7: $tType,O: $tType,Uu: fun(O,set(P7)),Uua: set(O)] : aa(set(O),set(P7),aTP_Lamp_ok(fun(O,set(P7)),fun(set(O),set(P7)),Uu),Uua) = aa(set(set(P7)),set(P7),complete_Inf_Inf(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),Uu),Uua)) ).

% ATP.lambda_319
tff(fact_8502_ATP_Olambda__320,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_ahz(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,Uu),Uua)) ) ).

% ATP.lambda_320
tff(fact_8503_ATP_Olambda__321,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_amx(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),Uu)) ) ).

% ATP.lambda_321
tff(fact_8504_ATP_Olambda__322,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(set(A),fun(fun(A,bool),bool),aTP_Lamp_ahk(set(A),fun(fun(A,bool),bool)),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Inf_Inf(bool),aa(set(A),set(bool),image2(A,bool,Uua),Uu))) ) ).

% ATP.lambda_322
tff(fact_8505_ATP_Olambda__323,axiom,
    ! [C: $tType,A: $tType] :
      ( complete_Inf(C)
     => ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_amw(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,Uua),Uu)) ) ).

% ATP.lambda_323
tff(fact_8506_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_bc(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_324
tff(fact_8507_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_bf(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_325
tff(fact_8508_ATP_Olambda__326,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_aqa(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_326
tff(fact_8509_ATP_Olambda__327,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_aqd(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(code_natural,A,pick(A,Uu),Uua)) ).

% ATP.lambda_327
tff(fact_8510_ATP_Olambda__328,axiom,
    ! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: A] : aa(A,B,aTP_Lamp_acl(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_328
tff(fact_8511_ATP_Olambda__329,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_sk(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))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_329
tff(fact_8512_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_yx(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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_330
tff(fact_8513_ATP_Olambda__331,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abg(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu),Uua)) ).

% ATP.lambda_331
tff(fact_8514_ATP_Olambda__332,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abf(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu)) ).

% ATP.lambda_332
tff(fact_8515_ATP_Olambda__333,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ado(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).

% ATP.lambda_333
tff(fact_8516_ATP_Olambda__334,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_adp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).

% ATP.lambda_334
tff(fact_8517_ATP_Olambda__335,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_tr(A,fun(A,bool),Uu),Uua))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uua)) ) ) ).

% ATP.lambda_335
tff(fact_8518_ATP_Olambda__336,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_cj(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,member(A,Uua),Uu)) ) ).

% ATP.lambda_336
tff(fact_8519_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_sn(A,fun(A,bool)),Uu),Uua))
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_337
tff(fact_8520_ATP_Olambda__338,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aon(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_338
tff(fact_8521_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aov(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_339
tff(fact_8522_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_sp(A,fun(A,bool)),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_340
tff(fact_8523_ATP_Olambda__341,axiom,
    ! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_zf(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)),insert(A),Uu),bot_bot(set(A)))) ).

% ATP.lambda_341
tff(fact_8524_ATP_Olambda__342,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_aps(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_342
tff(fact_8525_ATP_Olambda__343,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_acz(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_343
tff(fact_8526_ATP_Olambda__344,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_aqp(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_344
tff(fact_8527_ATP_Olambda__345,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,set(product_prod(C,C)),aTP_Lamp_ars(fun(B,set(C)),fun(B,set(product_prod(C,C))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Uu,Uua)) ).

% ATP.lambda_345
tff(fact_8528_ATP_Olambda__346,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_aro(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_346
tff(fact_8529_ATP_Olambda__347,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_ku(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_347
tff(fact_8530_ATP_Olambda__348,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [Uu: fun(B,rat),Uua: B] : aa(B,A,aTP_Lamp_ags(fun(B,rat),fun(B,A),Uu),Uua) = aa(rat,A,field_char_0_of_rat(A),aa(B,rat,Uu,Uua)) ) ).

% ATP.lambda_348
tff(fact_8531_ATP_Olambda__349,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_dp(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).

% ATP.lambda_349
tff(fact_8532_ATP_Olambda__350,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_eh(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_350
tff(fact_8533_ATP_Olambda__351,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_da(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_351
tff(fact_8534_ATP_Olambda__352,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_nt(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_352
tff(fact_8535_ATP_Olambda__353,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_on(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_353
tff(fact_8536_ATP_Olambda__354,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ea(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_8537_ATP_Olambda__355,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_zc(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_355
tff(fact_8538_ATP_Olambda__356,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_356
tff(fact_8539_ATP_Olambda__357,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ei(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_357
tff(fact_8540_ATP_Olambda__358,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_db(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_358
tff(fact_8541_ATP_Olambda__359,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_qp(fun(B,A),fun(B,heap_Time_Heap(A)),Uu),Uua) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_359
tff(fact_8542_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_di(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_360
tff(fact_8543_ATP_Olambda__361,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_eq(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_8544_ATP_Olambda__362,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_dm(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_362
tff(fact_8545_ATP_Olambda__363,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_ob(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_363
tff(fact_8546_ATP_Olambda__364,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_qx(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_364
tff(fact_8547_ATP_Olambda__365,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_lz(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_365
tff(fact_8548_ATP_Olambda__366,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_afk(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_366
tff(fact_8549_ATP_Olambda__367,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: C] : aa(C,set(product_prod(A,B)),aTP_Lamp_alc(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),Uu),Uua) = converse(B,A,aa(C,set(product_prod(B,A)),Uu,Uua)) ).

% ATP.lambda_367
tff(fact_8550_ATP_Olambda__368,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_abw(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).

% ATP.lambda_368
tff(fact_8551_ATP_Olambda__369,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_aem(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).

% ATP.lambda_369
tff(fact_8552_ATP_Olambda__370,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_ang(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_370
tff(fact_8553_ATP_Olambda__371,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_ank(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = principal(D,aa(C,set(D),Uu,Uua)) ).

% ATP.lambda_371
tff(fact_8554_ATP_Olambda__372,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_adm(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = principal(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_372
tff(fact_8555_ATP_Olambda__373,axiom,
    ! [E: $tType,A: $tType,Uu: fun(A,set(E)),Uua: A] : aa(A,filter(E),aTP_Lamp_anj(fun(A,set(E)),fun(A,filter(E)),Uu),Uua) = principal(E,aa(A,set(E),Uu,Uua)) ).

% ATP.lambda_373
tff(fact_8556_ATP_Olambda__374,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_adl(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_374
tff(fact_8557_ATP_Olambda__375,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_me(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_375
tff(fact_8558_ATP_Olambda__376,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(B),aTP_Lamp_arp(fun(A,set(product_prod(B,B))),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(B,B)),set(B),field2(B),aa(A,set(product_prod(B,B)),Uu,Uua)) ).

% ATP.lambda_376
tff(fact_8559_ATP_Olambda__377,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_rl(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_377
tff(fact_8560_ATP_Olambda__378,axiom,
    ! [A: $tType,Uu: fun(set(A),fun(A,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_aqn(fun(set(A),fun(A,bool)),fun(set(A),set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),Uu,Uua)) ).

% ATP.lambda_378
tff(fact_8561_ATP_Olambda__379,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: A] : aa(A,set(B),aTP_Lamp_yb(fun(A,fun(B,bool)),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),Uu,Uua)) ).

% ATP.lambda_379
tff(fact_8562_ATP_Olambda__380,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_st(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_380
tff(fact_8563_ATP_Olambda__381,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_tc(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_381
tff(fact_8564_ATP_Olambda__382,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_ld(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_382
tff(fact_8565_ATP_Olambda__383,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_cu(fun(B,bool),fun(B,bool),Uu),Uua))
    <=> ~ pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_383
tff(fact_8566_ATP_Olambda__384,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ci(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_384
tff(fact_8567_ATP_Olambda__385,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_aqj(fun(B,fun(A,bool)),fun(B,bool),Uu),Uua))
    <=> ! [X_12: A] : pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uua),X_12)) ) ).

% ATP.lambda_385
tff(fact_8568_ATP_Olambda__386,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(A,fun(B,bool)),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aop(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
        <=> ! [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_12)) ) ) ).

% ATP.lambda_386
tff(fact_8569_ATP_Olambda__387,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aiy(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ! [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_12)) ) ).

% ATP.lambda_387
tff(fact_8570_ATP_Olambda__388,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aaf(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_12)) ) ).

% ATP.lambda_388
tff(fact_8571_ATP_Olambda__389,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_arv(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cb(nat,fun(nat,bool)),Uu)) ).

% ATP.lambda_389
tff(fact_8572_ATP_Olambda__390,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_aea(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),bool),aTP_Lamp_adz(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua))) ).

% ATP.lambda_390
tff(fact_8573_ATP_Olambda__391,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aam(list(A),fun(A,bool),Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).

% ATP.lambda_391
tff(fact_8574_ATP_Olambda__392,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(option(A))),Uua: set(option(A))] :
          ( pp(aa(set(option(A)),bool,aTP_Lamp_aic(set(set(option(A))),fun(set(option(A)),bool),Uu),Uua))
        <=> ? [F10: fun(set(option(A)),option(A))] :
              ( ( Uua = aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),F10),Uu) )
              & ! [X4: set(option(A))] :
                  ( pp(aa(set(set(option(A))),bool,member(set(option(A)),X4),Uu))
                 => pp(aa(set(option(A)),bool,member(option(A),aa(set(option(A)),option(A),F10,X4)),X4)) ) ) ) ) ).

% ATP.lambda_392
tff(fact_8575_ATP_Olambda__393,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( pp(aa(set(B),bool,aTP_Lamp_aib(set(set(B)),fun(set(B),bool),Uu),Uua))
    <=> ? [F10: fun(set(B),B)] :
          ( ( Uua = aa(set(set(B)),set(B),image2(set(B),B,F10),Uu) )
          & ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X4),Uu))
             => pp(aa(set(B),bool,member(B,aa(set(B),B,F10,X4)),X4)) ) ) ) ).

% ATP.lambda_393
tff(fact_8576_ATP_Olambda__394,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_ahv(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F10: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F10),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F10,X4)),X4)) ) ) ) ) ).

% ATP.lambda_394
tff(fact_8577_ATP_Olambda__395,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_ahx(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F10: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F10),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F10,X4)),X4)) ) ) ) ) ).

% ATP.lambda_395
tff(fact_8578_ATP_Olambda__396,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_ahy(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F10: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F10),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F10,X4)),X4)) ) ) ) ) ).

% ATP.lambda_396
tff(fact_8579_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aab(set(A),fun(set(A),bool),Uu),Uua))
    <=> ? [B7: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B7) )
          & pp(aa(set(set(A)),bool,member(set(A),Uu),pow(A,B7))) ) ) ).

% ATP.lambda_397
tff(fact_8580_ATP_Olambda__398,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( pp(aa(filter(A),bool,aTP_Lamp_ahw(set(filter(A)),fun(filter(A),bool),Uu),Uua))
    <=> ! [X4: filter(A)] :
          ( pp(aa(set(filter(A)),bool,member(filter(A),X4),Uu))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),Uua),X4)) ) ) ).

% ATP.lambda_398
tff(fact_8581_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahp(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X4)) ) ) ) ).

% ATP.lambda_399
tff(fact_8582_ATP_Olambda__400,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahq(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Uua)) ) ) ) ).

% ATP.lambda_400
tff(fact_8583_ATP_Olambda__401,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_amg(set(multiset(A)),fun(multiset(A),bool),Uu),Uua))
    <=> ! [X4: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),Uu))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Uua),X4)) ) ) ).

% ATP.lambda_401
tff(fact_8584_ATP_Olambda__402,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_alw(set(multiset(A)),fun(multiset(A),bool),Uu),Uua))
    <=> ! [X4: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),Uu))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X4),Uua)) ) ) ).

% ATP.lambda_402
tff(fact_8585_ATP_Olambda__403,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aTP_Lamp_api(set(filter(A)),fun(fun(A,bool),bool),Uu),Uua))
    <=> ! [X4: filter(A)] :
          ( pp(aa(set(filter(A)),bool,member(filter(A),X4),Uu))
         => eventually(A,Uua,X4) ) ) ).

% ATP.lambda_403
tff(fact_8586_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ahs(set(set(A)),fun(A,bool),Uu),Uua))
    <=> ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
         => pp(aa(set(A),bool,member(A,Uua),X4)) ) ) ).

% ATP.lambda_404
tff(fact_8587_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aip(set(set(A)),fun(A,bool),Uu),Uua))
    <=> ? [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
          & pp(aa(set(A),bool,member(A,Uua),X4)) ) ) ).

% ATP.lambda_405
tff(fact_8588_ATP_Olambda__406,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aoy(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5))
             => pp(aa(A,bool,Uu,Y5)) ) ) ) ).

% ATP.lambda_406
tff(fact_8589_ATP_Olambda__407,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ana(fun(A,bool),fun(set(A),bool),Uu),Uua))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
         => pp(aa(A,bool,Uu,X4)) ) ) ).

% ATP.lambda_407
tff(fact_8590_ATP_Olambda__408,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ant(set(product_prod(A,A)),fun(set(A),bool),Uu),Uua))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),Uua))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),Uu))
                | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_408
tff(fact_8591_ATP_Olambda__409,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_aaa(fun(A,option(B)),fun(B,bool),Uu),Uua))
    <=> ? [A8: A] : aa(A,option(B),Uu,A8) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_409
tff(fact_8592_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_aal(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
    <=> ? [X4: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Uu,X4)),Uua)) ) ).

% ATP.lambda_410
tff(fact_8593_ATP_Olambda__411,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_ze(A,fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [V6: B] : Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),V6) ) ).

% ATP.lambda_411
tff(fact_8594_ATP_Olambda__412,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_zy(fun(B,A),fun(A,bool),Uu),Uua))
    <=> ? [X4: B] : Uua = aa(B,A,Uu,X4) ) ).

% ATP.lambda_412
tff(fact_8595_ATP_Olambda__413,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,bool),Uua: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aTP_Lamp_amm(fun(B,bool),fun(fun(A,B),bool),Uu),Uua))
    <=> ! [X4: A] : pp(aa(B,bool,Uu,aa(A,B,Uua,X4))) ) ).

% ATP.lambda_413
tff(fact_8596_ATP_Olambda__414,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: product_prod(product_prod(bool,A),product_prod(bool,B))] :
      ( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_aaq(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu),Uua))
    <=> ? [X4: A,Y5: B] :
          ( ( Uua = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fFalse),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y5)) )
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y5)),Uu)) ) ) ).

% ATP.lambda_414
tff(fact_8597_ATP_Olambda__415,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_aaj(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [A8: A,B13: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B13) )
          & ( aa(A,option(B),Uu,A8) = aa(B,option(B),some(B),B13) ) ) ) ).

% ATP.lambda_415
tff(fact_8598_ATP_Olambda__416,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( pp(aa(product_prod(set(A),set(A)),bool,aTP_Lamp_aih(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
    <=> ? [X14: set(A),Y9: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X14),Y9) )
          & ( X14 != bot_bot(set(A)) )
          & ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Y9))
             => ? [Xa: A] :
                  ( pp(aa(set(A),bool,member(A,Xa),X14))
                  & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_416
tff(fact_8599_ATP_Olambda__417,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_xv(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_417
tff(fact_8600_ATP_Olambda__418,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_abv(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ).

% ATP.lambda_418
tff(fact_8601_ATP_Olambda__419,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_we(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_419
tff(fact_8602_ATP_Olambda__420,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: C] : aa(C,set(B),aTP_Lamp_arn(set(product_prod(B,B)),fun(C,set(B)),Uu),Uua) = aa(set(product_prod(B,B)),set(B),field2(B),Uu) ).

% ATP.lambda_420
tff(fact_8603_ATP_Olambda__421,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_arr(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(B,B)),set(B),field2(B),Uu) ).

% ATP.lambda_421
tff(fact_8604_ATP_Olambda__422,axiom,
    ! [C: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: C] : aa(C,set(A),aTP_Lamp_arm(set(product_prod(A,A)),fun(C,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ).

% ATP.lambda_422
tff(fact_8605_ATP_Olambda__423,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_arq(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ).

% ATP.lambda_423
tff(fact_8606_ATP_Olambda__424,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_aje(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ).

% ATP.lambda_424
tff(fact_8607_ATP_Olambda__425,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_xy(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_425
tff(fact_8608_ATP_Olambda__426,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,set(B),aTP_Lamp_xr(fun(B,bool),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),Uu) ).

% ATP.lambda_426
tff(fact_8609_ATP_Olambda__427,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_ann(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert(A),Uu) ).

% ATP.lambda_427
tff(fact_8610_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_ajk(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_428
tff(fact_8611_ATP_Olambda__429,axiom,
    ! [A: $tType,Uu: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_ru(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).

% ATP.lambda_429
tff(fact_8612_ATP_Olambda__430,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_kk(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_430
tff(fact_8613_ATP_Olambda__431,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_iz(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_431
tff(fact_8614_ATP_Olambda__432,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ja(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_432
tff(fact_8615_ATP_Olambda__433,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_ir(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).

% ATP.lambda_433
tff(fact_8616_ATP_Olambda__434,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_tt(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),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_434
tff(fact_8617_ATP_Olambda__435,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_gw(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,member(B,Uub),Uua),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_435
tff(fact_8618_ATP_Olambda__436,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_hb(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,member(B,Uub),Uua),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_436
tff(fact_8619_ATP_Olambda__437,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fl(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_437
tff(fact_8620_ATP_Olambda__438,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fn(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_438
tff(fact_8621_ATP_Olambda__439,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fm(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_439
tff(fact_8622_ATP_Olambda__440,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fo(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_440
tff(fact_8623_ATP_Olambda__441,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_ame(A,fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uu),aa(nat,nat,suc,aa(A,nat,Uua,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_441
tff(fact_8624_ATP_Olambda__442,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] : aa(A,nat,aa(A,fun(A,nat),aTP_Lamp_amq(fun(A,nat),fun(A,fun(A,nat)),Uu),Uua),Uub) = if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua),aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_442
tff(fact_8625_ATP_Olambda__443,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_alr(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),Uua,zero_zero(A)) ) ).

% ATP.lambda_443
tff(fact_8626_ATP_Olambda__444,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_apt(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),Uua,one_one(A)) ) ).

% ATP.lambda_444
tff(fact_8627_ATP_Olambda__445,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_als(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),Uua,zero_zero(A)) ) ).

% ATP.lambda_445
tff(fact_8628_ATP_Olambda__446,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_apu(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),Uua,one_one(A)) ) ).

% ATP.lambda_446
tff(fact_8629_ATP_Olambda__447,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_lv(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_447
tff(fact_8630_ATP_Olambda__448,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_oy(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_448
tff(fact_8631_ATP_Olambda__449,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_lu(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).

% ATP.lambda_449
tff(fact_8632_ATP_Olambda__450,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_adu(fun(A,bool),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,Uu,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).

% ATP.lambda_450
tff(fact_8633_ATP_Olambda__451,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_su(fun(A,bool),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = if(set(A),aa(A,bool,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),Uub),Uub) ).

% ATP.lambda_451
tff(fact_8634_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,bool),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_amn(fun(A,bool),fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = if(nat,aa(A,bool,Uu,Uub),aa(A,nat,Uua,Uub),zero_zero(nat)) ).

% ATP.lambda_452
tff(fact_8635_ATP_Olambda__453,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_gh(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_453
tff(fact_8636_ATP_Olambda__454,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_vc(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_454
tff(fact_8637_ATP_Olambda__455,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,bool),Uub: A] : aa(A,nat,aa(fun(A,bool),fun(A,nat),aTP_Lamp_and(fun(A,nat),fun(fun(A,bool),fun(A,nat)),Uu),Uua),Uub) = if(nat,aa(A,bool,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat)) ).

% ATP.lambda_455
tff(fact_8638_ATP_Olambda__456,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_gm(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_456
tff(fact_8639_ATP_Olambda__457,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: fun(B,bool),Uub: B] : aa(B,fun(A,A),aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_adh(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),Uu),Uua),Uub) = if(fun(A,A),aa(B,bool,Uua,Uub),aa(B,fun(A,A),Uu,Uub),id(A)) ).

% ATP.lambda_457
tff(fact_8640_ATP_Olambda__458,axiom,
    ! [A: $tType,Uu: fun(heap_ext(product_unit),bool),Uua: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_bt(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = if(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),bool,Uu,Uub),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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Uua,Uub)),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ).

% ATP.lambda_458
tff(fact_8641_ATP_Olambda__459,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_uz(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_459
tff(fact_8642_ATP_Olambda__460,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_yy(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_yx(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_460
tff(fact_8643_ATP_Olambda__461,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_xm(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_sk(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_461
tff(fact_8644_ATP_Olambda__462,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_bw(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,aa(A,option(A),some(A),Uub),aa(A,fun(A,option(A)),aTP_Lamp_bv(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub),Uua) ).

% ATP.lambda_462
tff(fact_8645_ATP_Olambda__463,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_xf(fun(A,fun(A,bool)),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_463
tff(fact_8646_ATP_Olambda__464,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: option(A),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_bd(option(A),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,Uu,aTP_Lamp_bc(A,fun(A,option(A)),Uub),Uua) ) ).

% ATP.lambda_464
tff(fact_8647_ATP_Olambda__465,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_abo(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).

% ATP.lambda_465
tff(fact_8648_ATP_Olambda__466,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aa(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_adw(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub) = fun_upd(A,option(B),Uub,Uu,aa(B,option(B),some(B),Uua)) ).

% ATP.lambda_466
tff(fact_8649_ATP_Olambda__467,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aez(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ).

% ATP.lambda_467
tff(fact_8650_ATP_Olambda__468,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: bool] : aa(bool,set(A),aa(set(A),fun(bool,set(A)),aTP_Lamp_pt(set(A),fun(set(A),fun(bool,set(A))),Uu),Uua),Uub) = if(set(A),Uub,Uu,Uua) ).

% ATP.lambda_468
tff(fact_8651_ATP_Olambda__469,axiom,
    ! [B: $tType,A: $tType,Uu: heap_Time_Heap(A),Uua: fun(A,heap_Time_Heap(B)),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_jp(heap_Time_Heap(A),fun(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = case_option(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),none(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(A,product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_jo(fun(A,heap_Time_Heap(B)),fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uua)),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Uu),Uub)) ).

% ATP.lambda_469
tff(fact_8652_ATP_Olambda__470,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: product_prod(D,B)] : aa(product_prod(D,B),C,aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_wv(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,aa(product_prod(D,B),D,product_fst(D,B),Uub))),aa(product_prod(D,B),B,product_snd(D,B),Uub)) ).

% ATP.lambda_470
tff(fact_8653_ATP_Olambda__471,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pf(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ).

% ATP.lambda_471
tff(fact_8654_ATP_Olambda__472,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_ji(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_472
tff(fact_8655_ATP_Olambda__473,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_jg(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_473
tff(fact_8656_ATP_Olambda__474,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_aok(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_474
tff(fact_8657_ATP_Olambda__475,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_ho(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_475
tff(fact_8658_ATP_Olambda__476,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_hk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_476
tff(fact_8659_ATP_Olambda__477,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_pd(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_477
tff(fact_8660_ATP_Olambda__478,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_mn(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_478
tff(fact_8661_ATP_Olambda__479,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_ns(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_479
tff(fact_8662_ATP_Olambda__480,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_ade(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ).

% ATP.lambda_480
tff(fact_8663_ATP_Olambda__481,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_mt(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_481
tff(fact_8664_ATP_Olambda__482,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_ek(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_482
tff(fact_8665_ATP_Olambda__483,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_ds(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_483
tff(fact_8666_ATP_Olambda__484,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_zh(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_484
tff(fact_8667_ATP_Olambda__485,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_adi(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_485
tff(fact_8668_ATP_Olambda__486,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(A,fun(B,bool)),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aoo(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ) ).

% ATP.lambda_486
tff(fact_8669_ATP_Olambda__487,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aag(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_487
tff(fact_8670_ATP_Olambda__488,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_xk(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_488
tff(fact_8671_ATP_Olambda__489,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_adj(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_489
tff(fact_8672_ATP_Olambda__490,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_490
tff(fact_8673_ATP_Olambda__491,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_lo(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),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))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).

% ATP.lambda_491
tff(fact_8674_ATP_Olambda__492,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bb(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uua),Uub))
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_492
tff(fact_8675_ATP_Olambda__493,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_hp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ho(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_493
tff(fact_8676_ATP_Olambda__494,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_hl(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_494
tff(fact_8677_ATP_Olambda__495,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_apx(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_apw(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_495
tff(fact_8678_ATP_Olambda__496,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agf(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_age(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_496
tff(fact_8679_ATP_Olambda__497,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agd(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_agc(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_497
tff(fact_8680_ATP_Olambda__498,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_aga(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_afz(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_498
tff(fact_8681_ATP_Olambda__499,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_afy(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_afx(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_499
tff(fact_8682_ATP_Olambda__500,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_afw(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_afv(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_500
tff(fact_8683_ATP_Olambda__501,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_afu(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_aft(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_501
tff(fact_8684_ATP_Olambda__502,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_el(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(C,fun(B,A),aTP_Lamp_ek(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_502
tff(fact_8685_ATP_Olambda__503,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_dt(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(C,fun(B,A),aTP_Lamp_ds(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_503
tff(fact_8686_ATP_Olambda__504,axiom,
    ! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(B,C),Uub: product_prod(D,E)] : aa(product_prod(D,E),set(A),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_ow(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),set(A))),Uu),Uua),Uub) = aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(product_prod(D,E),fun(B,fun(C,set(A))),aTP_Lamp_ov(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uub)),Uua) ).

% ATP.lambda_504
tff(fact_8687_ATP_Olambda__505,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_fk(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_505
tff(fact_8688_ATP_Olambda__506,axiom,
    ! [A: $tType,Uu: fun(set(A),set(A)),Uua: set(A),Uub: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_aqq(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),Uu,Uub)),Uua)),complete_lattice_gfp(set(A),Uu)) ).

% ATP.lambda_506
tff(fact_8689_ATP_Olambda__507,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A)] : aa(set(A),set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_aql(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),Uua,Uub)),Uu)),complete_lattice_gfp(set(A),Uua)) ).

% ATP.lambda_507
tff(fact_8690_ATP_Olambda__508,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_vh(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_508
tff(fact_8691_ATP_Olambda__509,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_uy(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub)))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & pp(aa(set(product_prod(list(A),list(A))),bool,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_509
tff(fact_8692_ATP_Olambda__510,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_aax(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_510
tff(fact_8693_ATP_Olambda__511,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_aae(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X4: A,Y5: A,Xs6: 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),X4),Xs6)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),Uu)) ) ) ) ).

% ATP.lambda_511
tff(fact_8694_ATP_Olambda__512,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_vi(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_512
tff(fact_8695_ATP_Olambda__513,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_lr(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).

% ATP.lambda_513
tff(fact_8696_ATP_Olambda__514,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_lq(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).

% ATP.lambda_514
tff(fact_8697_ATP_Olambda__515,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_yz(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_515
tff(fact_8698_ATP_Olambda__516,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_rx(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).

% ATP.lambda_516
tff(fact_8699_ATP_Olambda__517,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_li(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).

% ATP.lambda_517
tff(fact_8700_ATP_Olambda__518,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ln(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_518
tff(fact_8701_ATP_Olambda__519,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_vg(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_519
tff(fact_8702_ATP_Olambda__520,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,option(B))] :
      ( pp(aa(fun(A,option(B)),bool,aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_afn(set(A),fun(set(B),fun(fun(A,option(B)),bool)),Uu),Uua),Uub))
    <=> ( ( dom(A,B,Uub) = Uu )
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),ran(A,B,Uub)),Uua)) ) ) ).

% ATP.lambda_520
tff(fact_8703_ATP_Olambda__521,axiom,
    ! [Uu: heap_ext(product_unit),Uua: heap_ext(product_unit),Uub: nat] :
      ( pp(aa(nat,bool,aa(heap_ext(product_unit),fun(nat,bool),aTP_Lamp_aj(heap_ext(product_unit),fun(heap_ext(product_unit),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,Uu)),Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),lim(product_unit,Uua))) ) ) ).

% ATP.lambda_521
tff(fact_8704_ATP_Olambda__522,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lb(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,aa(nat,nat,suc,Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_522
tff(fact_8705_ATP_Olambda__523,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_acc(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua)),Uu)) ) ).

% ATP.lambda_523
tff(fact_8706_ATP_Olambda__524,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_uj(set(list(A)),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A)))),Uu)) ) ).

% ATP.lambda_524
tff(fact_8707_ATP_Olambda__525,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: fun(A,A),Uua: ref(A),Uub: A] : aa(A,heap_Time_Heap(A),aa(ref(A),fun(A,heap_Time_Heap(A)),aTP_Lamp_ro(fun(A,A),fun(ref(A),fun(A,heap_Time_Heap(A))),Uu),Uua),Uub) = heap_Time_bind(product_unit,A,ref_update(A,Uua,aa(A,A,Uu,Uub)),aa(A,fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_rn(fun(A,A),fun(A,fun(product_unit,heap_Time_Heap(A))),Uu),Uub)) ) ).

% ATP.lambda_525
tff(fact_8708_ATP_Olambda__526,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_td(nat,fun(nat,fun(set(nat),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(nat)),bool,member(set(nat),Uub),pow(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu))))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_526
tff(fact_8709_ATP_Olambda__527,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ach(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
        & pp(aa(set(nat),bool,member(nat,Uub),Uua)) ) ) ).

% ATP.lambda_527
tff(fact_8710_ATP_Olambda__528,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_ss(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
        & pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).

% ATP.lambda_528
tff(fact_8711_ATP_Olambda__529,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: multiset(A),Uub: A] :
      ( pp(aa(A,bool,aa(multiset(A),fun(A,bool),aTP_Lamp_ane(fun(A,bool),fun(multiset(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(multiset(A),set(A),set_mset(A),Uua)))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_529
tff(fact_8712_ATP_Olambda__530,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aej(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),aa(list(A),set(A),set2(A),Uu)))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_530
tff(fact_8713_ATP_Olambda__531,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_sm(fun(A,bool),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(list(A),set(A),set2(A),Uua)))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_531
tff(fact_8714_ATP_Olambda__532,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(A,fun(product_prod(A,B),bool),aTP_Lamp_wo(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Uub),graph(A,B,Uu)))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_532
tff(fact_8715_ATP_Olambda__533,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hy(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_533
tff(fact_8716_ATP_Olambda__534,axiom,
    ! [Uu: heap_ext(product_unit),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ak(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),lim(product_unit,Uu)))
        & ~ pp(aa(set(nat),bool,member(nat,Uub),Uua)) ) ) ).

% ATP.lambda_534
tff(fact_8717_ATP_Olambda__535,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ajm(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu)) ) ) ) ).

% ATP.lambda_535
tff(fact_8718_ATP_Olambda__536,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ajn(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu)) ) ) ) ).

% ATP.lambda_536
tff(fact_8719_ATP_Olambda__537,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ajl(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => ( ( Uub != X4 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_537
tff(fact_8720_ATP_Olambda__538,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aqr(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => ( ( Uub != X4 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu)) ) ) ) ) ).

% ATP.lambda_538
tff(fact_8721_ATP_Olambda__539,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_acf(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))),Uua)) ) ).

% ATP.lambda_539
tff(fact_8722_ATP_Olambda__540,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ajh(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).

% ATP.lambda_540
tff(fact_8723_ATP_Olambda__541,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_akg(set(A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,Uu,Uua)),Uub)) ) ) ) ).

% ATP.lambda_541
tff(fact_8724_ATP_Olambda__542,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_aqx(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),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) )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub)) ) ) ).

% ATP.lambda_542
tff(fact_8725_ATP_Olambda__543,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ait(set(nat),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,Uub),Uu))
        & pp(aa(set(nat),bool,member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ais(set(nat),fun(nat,fun(nat,bool)),Uu),Uub)))),Uua)) ) ) ).

% ATP.lambda_543
tff(fact_8726_ATP_Olambda__544,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(nat,fun(set(A),bool),aTP_Lamp_kx(set(A),fun(nat,fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu))
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_544
tff(fact_8727_ATP_Olambda__545,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lc(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(nat,nat,suc,Uua))) ) ) ).

% ATP.lambda_545
tff(fact_8728_ATP_Olambda__546,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A),Uub: A] :
      ( pp(aa(A,bool,aa(multiset(A),fun(A,bool),aTP_Lamp_alm(multiset(A),fun(multiset(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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_546
tff(fact_8729_ATP_Olambda__547,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_ajv(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_547
tff(fact_8730_ATP_Olambda__548,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hg(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_548
tff(fact_8731_ATP_Olambda__549,axiom,
    ! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_bi(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uu),Uub))
        | pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uua),Uub)) ) ) ).

% ATP.lambda_549
tff(fact_8732_ATP_Olambda__550,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ah(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        | pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_550
tff(fact_8733_ATP_Olambda__551,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_qb(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_551
tff(fact_8734_ATP_Olambda__552,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jv(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_552
tff(fact_8735_ATP_Olambda__553,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ia(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_553
tff(fact_8736_ATP_Olambda__554,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ju(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_554
tff(fact_8737_ATP_Olambda__555,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_id(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_555
tff(fact_8738_ATP_Olambda__556,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ic(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_556
tff(fact_8739_ATP_Olambda__557,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ib(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_557
tff(fact_8740_ATP_Olambda__558,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ajt(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_558
tff(fact_8741_ATP_Olambda__559,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ajs(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uua))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).

% ATP.lambda_559
tff(fact_8742_ATP_Olambda__560,axiom,
    ! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_bj(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uu),Uub))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uua),Uub)) ) ) ).

% ATP.lambda_560
tff(fact_8743_ATP_Olambda__561,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_agh(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uu)) ) ) ).

% ATP.lambda_561
tff(fact_8744_ATP_Olambda__562,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_afr(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uu)) ) ) ).

% ATP.lambda_562
tff(fact_8745_ATP_Olambda__563,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_apk(filter(A),fun(filter(A),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ( eventually(A,Uub,Uu)
        & eventually(A,Uub,Uua) ) ) ).

% ATP.lambda_563
tff(fact_8746_ATP_Olambda__564,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aiu(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uua),Uu))
        & pp(aa(set(A),bool,member(A,Uub),Uu)) ) ) ).

% ATP.lambda_564
tff(fact_8747_ATP_Olambda__565,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ais(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_565
tff(fact_8748_ATP_Olambda__566,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_af(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_566
tff(fact_8749_ATP_Olambda__567,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_fq(set(A),fun(set(A),fun(set(A),bool)),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_567
tff(fact_8750_ATP_Olambda__568,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_vj(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_568
tff(fact_8751_ATP_Olambda__569,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_vf(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_569
tff(fact_8752_ATP_Olambda__570,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aox(fun(A,bool),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uua))
       => pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_570
tff(fact_8753_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: multiset(A),Uub: A] : aa(A,fun(B,B),aa(multiset(A),fun(A,fun(B,B)),aTP_Lamp_ami(fun(A,fun(B,B)),fun(multiset(A),fun(A,fun(B,B))),Uu),Uua),Uub) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_571
tff(fact_8754_ATP_Olambda__572,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_gg(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_572
tff(fact_8755_ATP_Olambda__573,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_akj(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_573
tff(fact_8756_ATP_Olambda__574,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ac(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_574
tff(fact_8757_ATP_Olambda__575,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_sv(fun(A,bool),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uua))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_575
tff(fact_8758_ATP_Olambda__576,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ui(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_576
tff(fact_8759_ATP_Olambda__577,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qe(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_577
tff(fact_8760_ATP_Olambda__578,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qd(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_578
tff(fact_8761_ATP_Olambda__579,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_yn(set(A),fun(fun(A,set(B)),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_579
tff(fact_8762_ATP_Olambda__580,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_gc(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
            & ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_580
tff(fact_8763_ATP_Olambda__581,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_qq(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_581
tff(fact_8764_ATP_Olambda__582,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ge(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
            & ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_582
tff(fact_8765_ATP_Olambda__583,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_qr(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uua))
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_583
tff(fact_8766_ATP_Olambda__584,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_rv(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uua))
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_584
tff(fact_8767_ATP_Olambda__585,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(B,A,Uua,Uub))) ) ) ) ).

% ATP.lambda_585
tff(fact_8768_ATP_Olambda__586,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_cf(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & ~ pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_586
tff(fact_8769_ATP_Olambda__587,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: multiset(A),Uub: A] : aa(A,nat,aa(multiset(A),fun(A,nat),aTP_Lamp_alu(fun(A,nat),fun(multiset(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(nat,nat,suc,aa(A,nat,Uu,Uub))) ).

% ATP.lambda_587
tff(fact_8770_ATP_Olambda__588,axiom,
    ! [A: $tType,D: $tType,C: $tType,Uu: fun(C,fun(D,bool)),Uua: fun(C,A),Uub: set(A)] : aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),aTP_Lamp_aod(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uua),Uub)),aa(fun(C,bool),set(C),collect(C),aa(fun(C,fun(D,bool)),fun(C,bool),domainp(C,D),Uu))) ).

% ATP.lambda_588
tff(fact_8771_ATP_Olambda__589,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_wq(list(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_589
tff(fact_8772_ATP_Olambda__590,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_sd(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uu)),Uua)) ) ).

% ATP.lambda_590
tff(fact_8773_ATP_Olambda__591,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jf(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_591
tff(fact_8774_ATP_Olambda__592,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jy(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_592
tff(fact_8775_ATP_Olambda__593,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_cl(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_593
tff(fact_8776_ATP_Olambda__594,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_cn(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_594
tff(fact_8777_ATP_Olambda__595,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_cm(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_595
tff(fact_8778_ATP_Olambda__596,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_co(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_596
tff(fact_8779_ATP_Olambda__597,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
      ( pp(aa(C,bool,aa(B,fun(C,bool),aTP_Lamp_aqw(set(product_prod(B,C)),fun(B,fun(C,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)),Uu)) ) ).

% ATP.lambda_597
tff(fact_8780_ATP_Olambda__598,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_bq(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_598
tff(fact_8781_ATP_Olambda__599,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ad(set(product_prod(A,B)),fun(A,fun(B,bool))),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu)) ) ).

% ATP.lambda_599
tff(fact_8782_ATP_Olambda__600,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_xi(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_600
tff(fact_8783_ATP_Olambda__601,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_alb(set(product_prod(B,A)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua)),Uu)) ) ).

% ATP.lambda_601
tff(fact_8784_ATP_Olambda__602,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_alg(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ).

% ATP.lambda_602
tff(fact_8785_ATP_Olambda__603,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vr(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_603
tff(fact_8786_ATP_Olambda__604,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_go(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uub))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uub),Uua)) ) ) ) ).

% ATP.lambda_604
tff(fact_8787_ATP_Olambda__605,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_th(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_605
tff(fact_8788_ATP_Olambda__606,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_fs(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,Uu,Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_606
tff(fact_8789_ATP_Olambda__607,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aoe(fun(A,fun(B,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu),Uub)) ) ) ).

% ATP.lambda_607
tff(fact_8790_ATP_Olambda__608,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_tf(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ) ).

% ATP.lambda_608
tff(fact_8791_ATP_Olambda__609,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_tg(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% ATP.lambda_609
tff(fact_8792_ATP_Olambda__610,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_yt(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_610
tff(fact_8793_ATP_Olambda__611,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_ys(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_611
tff(fact_8794_ATP_Olambda__612,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_us(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_612
tff(fact_8795_ATP_Olambda__613,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_apa(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_613
tff(fact_8796_ATP_Olambda__614,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eo(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_614
tff(fact_8797_ATP_Olambda__615,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,B)),Uua: fun(A,option(C)),Uub: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aTP_Lamp_afl(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),Uu),Uua),Uub) = aa(option(C),option(B),map_option(C,B,aa(A,fun(C,B),Uu,Uub)),aa(A,option(C),Uua,Uub)) ).

% ATP.lambda_615
tff(fact_8798_ATP_Olambda__616,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_en(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_616
tff(fact_8799_ATP_Olambda__617,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_dn(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_617
tff(fact_8800_ATP_Olambda__618,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_dz(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_618
tff(fact_8801_ATP_Olambda__619,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_yc(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_619
tff(fact_8802_ATP_Olambda__620,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_amd(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_620
tff(fact_8803_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_kc(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_621
tff(fact_8804_ATP_Olambda__622,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_dq(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_622
tff(fact_8805_ATP_Olambda__623,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ni(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_623
tff(fact_8806_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_mz(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_624
tff(fact_8807_ATP_Olambda__625,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_as(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_625
tff(fact_8808_ATP_Olambda__626,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_yf(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_626
tff(fact_8809_ATP_Olambda__627,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_od(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_627
tff(fact_8810_ATP_Olambda__628,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oc(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_628
tff(fact_8811_ATP_Olambda__629,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_yd(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_629
tff(fact_8812_ATP_Olambda__630,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_du(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_630
tff(fact_8813_ATP_Olambda__631,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_amc(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_631
tff(fact_8814_ATP_Olambda__632,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_aml(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(B,multiset(A),Uu,Uub)),aa(B,multiset(A),Uua,Uub)) ).

% ATP.lambda_632
tff(fact_8815_ATP_Olambda__633,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_amf(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(B,multiset(A),Uu,Uub)),aa(B,multiset(A),Uua,Uub)) ).

% ATP.lambda_633
tff(fact_8816_ATP_Olambda__634,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_acm(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_634
tff(fact_8817_ATP_Olambda__635,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_cg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_635
tff(fact_8818_ATP_Olambda__636,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_anq(fun(A,set(B)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),aa(A,set(B),Uu,Uua)),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_636
tff(fact_8819_ATP_Olambda__637,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: fun(A,nat),Uub: A] : aa(A,fun(B,B),aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_uk(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),Uu),Uua),Uub) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(A,nat,Uua,Uub)),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_637
tff(fact_8820_ATP_Olambda__638,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aqk(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(A,bool,Uu,Uub))
            | pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_638
tff(fact_8821_ATP_Olambda__639,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ai(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_639
tff(fact_8822_ATP_Olambda__640,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_wb(fun(B,bool),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(B,bool,Uu,Uub))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_640
tff(fact_8823_ATP_Olambda__641,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ag(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_641
tff(fact_8824_ATP_Olambda__642,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_sl(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_642
tff(fact_8825_ATP_Olambda__643,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_aci(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uua) = aa(B,A,Uu,Uub) ) ) ) ).

% ATP.lambda_643
tff(fact_8826_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aew(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_644
tff(fact_8827_ATP_Olambda__645,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aom(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_645
tff(fact_8828_ATP_Olambda__646,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aor(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
      <=> pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_646
tff(fact_8829_ATP_Olambda__647,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_anz(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_647
tff(fact_8830_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ul(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_648
tff(fact_8831_ATP_Olambda__649,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_is(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uua,Uub))) ) ).

% ATP.lambda_649
tff(fact_8832_ATP_Olambda__650,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajc(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & ! [Y5: A] :
            ( pp(aa(A,bool,Uua,Y5))
           => pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y5)) ) ) ) ).

% ATP.lambda_650
tff(fact_8833_ATP_Olambda__651,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_ar(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_651
tff(fact_8834_ATP_Olambda__652,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_jm(fun(A,option(B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(A,option(B),Uu,Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_652
tff(fact_8835_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_ej(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_653
tff(fact_8836_ATP_Olambda__654,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_dr(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_654
tff(fact_8837_ATP_Olambda__655,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ft(fun(nat,nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua)) ) ).

% ATP.lambda_655
tff(fact_8838_ATP_Olambda__656,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ma(fun(B,set(A)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_656
tff(fact_8839_ATP_Olambda__657,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_apf(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_657
tff(fact_8840_ATP_Olambda__658,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_apc(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_658
tff(fact_8841_ATP_Olambda__659,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_dc(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_659
tff(fact_8842_ATP_Olambda__660,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_in(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_660
tff(fact_8843_ATP_Olambda__661,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_eb(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_661
tff(fact_8844_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_662
tff(fact_8845_ATP_Olambda__663,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_apd(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_663
tff(fact_8846_ATP_Olambda__664,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_dx(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_664
tff(fact_8847_ATP_Olambda__665,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_aq(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_665
tff(fact_8848_ATP_Olambda__666,axiom,
    ! [K6: $tType,L5: $tType,Uu: fun(K6,set(L5)),Uua: set(L5),Uub: K6] : aa(K6,set(L5),aa(set(L5),fun(K6,set(L5)),aTP_Lamp_nj(fun(K6,set(L5)),fun(set(L5),fun(K6,set(L5))),Uu),Uua),Uub) = aa(set(L5),set(L5),aa(set(L5),fun(set(L5),set(L5)),minus_minus(set(L5)),aa(K6,set(L5),Uu,Uub)),Uua) ).

% ATP.lambda_666
tff(fact_8849_ATP_Olambda__667,axiom,
    ! [E: $tType,F: $tType,Uu: fun(E,set(F)),Uua: set(F),Uub: E] : aa(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_pb(fun(E,set(F)),fun(set(F),fun(E,set(F))),Uu),Uua),Uub) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(E,set(F),Uu,Uub)),Uua) ).

% ATP.lambda_667
tff(fact_8850_ATP_Olambda__668,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_ep(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_668
tff(fact_8851_ATP_Olambda__669,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,multiset(A)),Uua: A,Uub: B] : aa(B,nat,aa(A,fun(B,nat),aTP_Lamp_alk(fun(B,multiset(A)),fun(A,fun(B,nat)),Uu),Uua),Uub) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(B,multiset(A),Uu,Uub)),Uua) ).

% ATP.lambda_669
tff(fact_8852_ATP_Olambda__670,axiom,
    ! [K6: $tType,L5: $tType,Uu: fun(K6,set(L5)),Uua: set(L5),Uub: K6] : aa(K6,set(L5),aa(set(L5),fun(K6,set(L5)),aTP_Lamp_oh(fun(K6,set(L5)),fun(set(L5),fun(K6,set(L5))),Uu),Uua),Uub) = aa(set(L5),set(L5),aa(set(L5),fun(set(L5),set(L5)),sup_sup(set(L5)),aa(K6,set(L5),Uu,Uub)),Uua) ).

% ATP.lambda_670
tff(fact_8853_ATP_Olambda__671,axiom,
    ! [C: $tType,D: $tType,Uu: fun(C,set(D)),Uua: set(D),Uub: C] : aa(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_mg(fun(C,set(D)),fun(set(D),fun(C,set(D))),Uu),Uua),Uub) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(C,set(D),Uu,Uub)),Uua) ).

% ATP.lambda_671
tff(fact_8854_ATP_Olambda__672,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_oa(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_672
tff(fact_8855_ATP_Olambda__673,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_at(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_673
tff(fact_8856_ATP_Olambda__674,axiom,
    ! [G: $tType,H10: $tType,Uu: fun(G,set(H10)),Uua: set(H10),Uub: G] : aa(G,set(H10),aa(set(H10),fun(G,set(H10)),aTP_Lamp_ne(fun(G,set(H10)),fun(set(H10),fun(G,set(H10))),Uu),Uua),Uub) = aa(set(H10),set(H10),aa(set(H10),fun(set(H10),set(H10)),inf_inf(set(H10)),aa(G,set(H10),Uu,Uub)),Uua) ).

% ATP.lambda_674
tff(fact_8857_ATP_Olambda__675,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_nn(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_675
tff(fact_8858_ATP_Olambda__676,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_om(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_676
tff(fact_8859_ATP_Olambda__677,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_au(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_677
tff(fact_8860_ATP_Olambda__678,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_op(fun(A,set(B)),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),Uua) ).

% ATP.lambda_678
tff(fact_8861_ATP_Olambda__679,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_679
tff(fact_8862_ATP_Olambda__680,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,heap_Time_Heap(C)),Uua: fun(C,heap_Time_Heap(B)),Uub: A] : aa(A,heap_Time_Heap(B),aa(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aTP_Lamp_bl(fun(A,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),Uu),Uua),Uub) = heap_Time_bind(C,B,aa(A,heap_Time_Heap(C),Uu,Uub),Uua) ).

% ATP.lambda_680
tff(fact_8863_ATP_Olambda__681,axiom,
    ! [E: $tType,F: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F,B),aa(fun(C,fun(D,B)),fun(E,fun(F,B)),aTP_Lamp_aqb(fun(E,fun(F,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F,B))),Uu),Uua),Uub) = product_scomp(F,C,D,B,aa(E,fun(F,product_prod(C,D)),Uu,Uub),Uua) ).

% ATP.lambda_681
tff(fact_8864_ATP_Olambda__682,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,fun(A,fun(C,bool))),Uua: fun(C,fun(B,bool)),Uub: D] : aa(D,fun(A,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool))),aTP_Lamp_aqv(fun(D,fun(A,fun(C,bool))),fun(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool)))),Uu),Uua),Uub) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(D,fun(A,fun(C,bool)),Uu,Uub)),Uua) ).

% ATP.lambda_682
tff(fact_8865_ATP_Olambda__683,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(A,C))),Uua: set(product_prod(C,B)),Uub: D] : aa(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_xb(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,aa(D,set(product_prod(A,C)),Uu,Uub),Uua) ).

% ATP.lambda_683
tff(fact_8866_ATP_Olambda__684,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: set(B),Uub: C] : aa(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_ahe(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,aa(C,set(product_prod(B,A)),Uu,Uub)),Uua) ).

% ATP.lambda_684
tff(fact_8867_ATP_Olambda__685,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_acs(fun(A,B),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(B),bool,member(B,aa(A,B,Uu,Uub)),Uua)) ) ).

% ATP.lambda_685
tff(fact_8868_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aes(set(A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,aa(B,A,Uua,Uub)),Uu)) ) ).

% ATP.lambda_686
tff(fact_8869_ATP_Olambda__687,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: bool,Uub: A] :
      ( pp(aa(A,bool,aa(bool,fun(A,bool),aTP_Lamp_aos(fun(A,bool),fun(bool,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(Uua) ) ) ).

% ATP.lambda_687
tff(fact_8870_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tx(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_688
tff(fact_8871_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_689
tff(fact_8872_ATP_Olambda__690,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_afd(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_690
tff(fact_8873_ATP_Olambda__691,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_arc(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uua )
        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ) ).

% ATP.lambda_691
tff(fact_8874_ATP_Olambda__692,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qc(A,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_692
tff(fact_8875_ATP_Olambda__693,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gp(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_693
tff(fact_8876_ATP_Olambda__694,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ajg(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua != Uu )
        & ( Uub != Uu ) ) ) ).

% ATP.lambda_694
tff(fact_8877_ATP_Olambda__695,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_it(fun(B,bool),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uub))),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_695
tff(fact_8878_ATP_Olambda__696,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aid(fun(A,A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( ? [X4: A] :
                ( ( Uub = aa(A,A,Uu,X4) )
                & pp(aa(A,bool,Uua,X4)) )
            | ? [M13: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M13) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M13)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),M13))
                   => pp(aa(A,bool,Uua,X4)) ) ) ) ) ) ).

% ATP.lambda_696
tff(fact_8879_ATP_Olambda__697,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(set(B),fun(fun(A,B),bool),aTP_Lamp_aiz(set(A),fun(set(B),fun(fun(A,B),bool)),Uu),Uua),Uub))
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uu))
           => pp(aa(set(B),bool,member(B,aa(A,B,Uub,X4)),Uua)) )
        & ! [A8: A] :
            ( ~ pp(aa(set(A),bool,member(A,A8),Uu))
           => ( aa(A,B,Uub,A8) = undefined(B) ) ) ) ) ).

% ATP.lambda_697
tff(fact_8880_ATP_Olambda__698,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),bool)),Uua: list(A),Uub: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_zr(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
        <=> ( ? [Y5: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) ) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs3),Ys4)) ) ) ) ) ).

% ATP.lambda_698
tff(fact_8881_ATP_Olambda__699,axiom,
    ! [A: $tType,Uu: fun(list(multiset(A)),fun(list(multiset(A)),bool)),Uua: list(multiset(A)),Uub: list(multiset(A))] :
      ( pp(aa(list(multiset(A)),bool,aa(list(multiset(A)),fun(list(multiset(A)),bool),aa(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool)),aTP_Lamp_acr(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))),Uu),Uua),Uub))
    <=> ( ? [Y5: multiset(A),Ys4: list(multiset(A))] :
            ( ( Uua = nil(multiset(A)) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y5),Ys4) ) )
        | ? [X4: multiset(A),Y5: multiset(A),Xs3: list(multiset(A)),Ys4: list(multiset(A))] :
            ( ( Uua = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),X4),Xs3) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y5),Ys4) )
            & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),X4),Y5)) )
        | ? [X4: multiset(A),Y5: multiset(A),Xs3: list(multiset(A)),Ys4: list(multiset(A))] :
            ( ( Uua = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),X4),Xs3) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y5),Ys4) )
            & ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),X4),Y5))
            & ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Y5),X4))
            & pp(aa(list(multiset(A)),bool,aa(list(multiset(A)),fun(list(multiset(A)),bool),Uu,Xs3),Ys4)) ) ) ) ).

% ATP.lambda_699
tff(fact_8882_ATP_Olambda__700,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_adz(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_700
tff(fact_8883_ATP_Olambda__701,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_aiw(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
        & pp(aa(set(A),bool,finite_finite2(A),Uub))
        & ( Uub != bot_bot(set(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => ? [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),Uub))
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),Uu)) ) ) ) ) ).

% ATP.lambda_701
tff(fact_8884_ATP_Olambda__702,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_ald(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),image(A,A,converse(A,A,Uu)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A)))) ).

% ATP.lambda_702
tff(fact_8885_ATP_Olambda__703,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] : aa(B,fun(list(B),list(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_tq(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua),Uub) = case_list(list(B),B,Uua,aa(B,fun(B,fun(list(B),list(B))),aa(list(B),fun(B,fun(B,fun(list(B),list(B)))),aTP_Lamp_tp(fun(B,A),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub)) ) ).

% ATP.lambda_703
tff(fact_8886_ATP_Olambda__704,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ke(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_704
tff(fact_8887_ATP_Olambda__705,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kd(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_705
tff(fact_8888_ATP_Olambda__706,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: list(B),Uub: A] : aa(A,list(B),aa(list(B),fun(A,list(B)),aTP_Lamp_uw(fun(A,B),fun(list(B),fun(A,list(B))),Uu),Uua),Uub) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uua),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,Uu,Uub)),nil(B))) ).

% ATP.lambda_706
tff(fact_8889_ATP_Olambda__707,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ht(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_707
tff(fact_8890_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hf(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_708
tff(fact_8891_ATP_Olambda__709,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aig(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A)))))) ) ).

% ATP.lambda_709
tff(fact_8892_ATP_Olambda__710,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_kj(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_710
tff(fact_8893_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_afp(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,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_711
tff(fact_8894_ATP_Olambda__712,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_afq(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,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_712
tff(fact_8895_ATP_Olambda__713,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_dh(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_713
tff(fact_8896_ATP_Olambda__714,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_he(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_714
tff(fact_8897_ATP_Olambda__715,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_abj(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_715
tff(fact_8898_ATP_Olambda__716,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_abk(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uub)) ).

% ATP.lambda_716
tff(fact_8899_ATP_Olambda__717,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_tb(A,fun(list(A),fun(nat,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),take(A,Uub,Uua)) ).

% ATP.lambda_717
tff(fact_8900_ATP_Olambda__718,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_vb(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_718
tff(fact_8901_ATP_Olambda__719,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_akl(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(B),bool,member(B,Uub),aa(set(A),set(B),image2(A,B,Uu),Uua))) ) ) ).

% ATP.lambda_719
tff(fact_8902_ATP_Olambda__720,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_md(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_720
tff(fact_8903_ATP_Olambda__721,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_mc(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_721
tff(fact_8904_ATP_Olambda__722,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ape(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_722
tff(fact_8905_ATP_Olambda__723,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aoz(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_723
tff(fact_8906_ATP_Olambda__724,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ty(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_724
tff(fact_8907_ATP_Olambda__725,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_apb(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_725
tff(fact_8908_ATP_Olambda__726,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,option(C)),Uub: B] : aa(B,option(A),aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_qw(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),Uu),Uua),Uub) = aa(option(C),option(A),map_option(C,A,Uu),aa(B,option(C),Uua,Uub)) ).

% ATP.lambda_726
tff(fact_8909_ATP_Olambda__727,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_amh(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_727
tff(fact_8910_ATP_Olambda__728,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_dy(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_728
tff(fact_8911_ATP_Olambda__729,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: nat,Uub: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_amk(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_729
tff(fact_8912_ATP_Olambda__730,axiom,
    ! [M10: $tType,N9: $tType,Uu: set(M10),Uua: fun(N9,set(M10)),Uub: N9] : aa(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_oq(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),Uu),Uua),Uub) = aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),minus_minus(set(M10)),Uu),aa(N9,set(M10),Uua,Uub)) ).

% ATP.lambda_730
tff(fact_8913_ATP_Olambda__731,axiom,
    ! [G: $tType,H10: $tType,Uu: set(G),Uua: fun(H10,set(G)),Uub: H10] : aa(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_or(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),Uu),Uua),Uub) = aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),Uu),aa(H10,set(G),Uua,Uub)) ).

% ATP.lambda_731
tff(fact_8914_ATP_Olambda__732,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_et(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_732
tff(fact_8915_ATP_Olambda__733,axiom,
    ! [M10: $tType,N9: $tType,Uu: set(M10),Uua: fun(N9,set(M10)),Uub: N9] : aa(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_oi(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),Uu),Uua),Uub) = aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),sup_sup(set(M10)),Uu),aa(N9,set(M10),Uua,Uub)) ).

% ATP.lambda_733
tff(fact_8916_ATP_Olambda__734,axiom,
    ! [E: $tType,F: $tType,Uu: set(E),Uua: fun(F,set(E)),Uub: F] : aa(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_mh(set(E),fun(fun(F,set(E)),fun(F,set(E))),Uu),Uua),Uub) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),Uu),aa(F,set(E),Uua,Uub)) ).

% ATP.lambda_734
tff(fact_8917_ATP_Olambda__735,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_og(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_735
tff(fact_8918_ATP_Olambda__736,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ny(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_736
tff(fact_8919_ATP_Olambda__737,axiom,
    ! [I6: $tType,J4: $tType,Uu: set(I6),Uua: fun(J4,set(I6)),Uub: J4] : aa(J4,set(I6),aa(fun(J4,set(I6)),fun(J4,set(I6)),aTP_Lamp_nd(set(I6),fun(fun(J4,set(I6)),fun(J4,set(I6))),Uu),Uua),Uub) = aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),inf_inf(set(I6)),Uu),aa(J4,set(I6),Uua,Uub)) ).

% ATP.lambda_737
tff(fact_8920_ATP_Olambda__738,axiom,
    ! [C: $tType,D: $tType,Uu: set(C),Uua: fun(D,set(C)),Uub: D] : aa(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_oo(set(C),fun(fun(D,set(C)),fun(D,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),Uu),aa(D,set(C),Uua,Uub)) ).

% ATP.lambda_738
tff(fact_8921_ATP_Olambda__739,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_nf(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_739
tff(fact_8922_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_np(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_740
tff(fact_8923_ATP_Olambda__741,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ol(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_741
tff(fact_8924_ATP_Olambda__742,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_io(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_742
tff(fact_8925_ATP_Olambda__743,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,filter(C)),Uub: B] : aa(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_aqh(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,C,Uu,aa(B,filter(C),Uua,Uub)) ).

% ATP.lambda_743
tff(fact_8926_ATP_Olambda__744,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aqf(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_744
tff(fact_8927_ATP_Olambda__745,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(D,fun(C,fun(B,bool))),Uub: D] : aa(D,fun(A,fun(B,bool)),aa(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool))),aTP_Lamp_aqu(fun(A,fun(C,bool)),fun(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool)))),Uu),Uua),Uub) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),Uu),aa(D,fun(C,fun(B,bool)),Uua,Uub)) ).

% ATP.lambda_745
tff(fact_8928_ATP_Olambda__746,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_wi(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).

% ATP.lambda_746
tff(fact_8929_ATP_Olambda__747,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: set(product_prod(A,C)),Uua: fun(D,set(product_prod(C,B))),Uub: D] : aa(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_xa(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,Uu,aa(D,set(product_prod(C,B)),Uua,Uub)) ).

% ATP.lambda_747
tff(fact_8930_ATP_Olambda__748,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ahc(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_748
tff(fact_8931_ATP_Olambda__749,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aou(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_749
tff(fact_8932_ATP_Olambda__750,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_act(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_750
tff(fact_8933_ATP_Olambda__751,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_mr(fun(B,set(A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,Uua))) ) ).

% ATP.lambda_751
tff(fact_8934_ATP_Olambda__752,axiom,
    ! [I6: $tType,J4: $tType,Uu: I6,Uua: fun(J4,set(I6)),Uub: J4] : aa(J4,set(I6),aa(fun(J4,set(I6)),fun(J4,set(I6)),aTP_Lamp_qh(I6,fun(fun(J4,set(I6)),fun(J4,set(I6))),Uu),Uua),Uub) = aa(set(I6),set(I6),aa(I6,fun(set(I6),set(I6)),insert(I6),Uu),aa(J4,set(I6),Uua,Uub)) ).

% ATP.lambda_752
tff(fact_8935_ATP_Olambda__753,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_qg(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)),insert(B),Uu),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_753
tff(fact_8936_ATP_Olambda__754,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_py(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)),insert(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_754
tff(fact_8937_ATP_Olambda__755,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_nb(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_755
tff(fact_8938_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_aey(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image2(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_756
tff(fact_8939_ATP_Olambda__757,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aot(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_757
tff(fact_8940_ATP_Olambda__758,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_air(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        | pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_758
tff(fact_8941_ATP_Olambda__759,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_ahn(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        & pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_759
tff(fact_8942_ATP_Olambda__760,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_so(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_760
tff(fact_8943_ATP_Olambda__761,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_aeu(fun(A,B),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_761
tff(fact_8944_ATP_Olambda__762,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fd(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_762
tff(fact_8945_ATP_Olambda__763,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_acg(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(list(A),set(A),set2(A),nths(A,Uu,Uua)))) ) ).

% ATP.lambda_763
tff(fact_8946_ATP_Olambda__764,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: A] : aa(A,set(A),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_ard(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uua),aa(set(product_prod(B,B)),set(B),field2(B),Uu)) ).

% ATP.lambda_764
tff(fact_8947_ATP_Olambda__765,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(A),Uua: set(C),Uub: fun(A,B)] : aa(fun(A,B),set(fun(A,C)),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_arj(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),Uu),Uua),Uub) = bNF_Wellorder_Func(A,C,Uu,Uua) ).

% ATP.lambda_765
tff(fact_8948_ATP_Olambda__766,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] : aa(A,set(A),aa(set(A),fun(A,set(A)),aTP_Lamp_abu(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_766
tff(fact_8949_ATP_Olambda__767,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ye(set(B),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu),Uua) ).

% ATP.lambda_767
tff(fact_8950_ATP_Olambda__768,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_arb(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).

% ATP.lambda_768
tff(fact_8951_ATP_Olambda__769,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_aja(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_above(A,Uu,Uua) ).

% ATP.lambda_769
tff(fact_8952_ATP_Olambda__770,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: A] : aa(A,set(sum_sum(B,C)),aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_aru(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Uu),Uua),Uub) = sum_Plus(B,C,Uu,Uua) ).

% ATP.lambda_770
tff(fact_8953_ATP_Olambda__771,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_vp(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_771
tff(fact_8954_ATP_Olambda__772,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_xz(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uu),Uua) ).

% ATP.lambda_772
tff(fact_8955_ATP_Olambda__773,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_ym(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uu),Uua) ).

% ATP.lambda_773
tff(fact_8956_ATP_Olambda__774,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_xx(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_774
tff(fact_8957_ATP_Olambda__775,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_gv(fun(A,nat),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua),aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub)))) ) ).

% ATP.lambda_775
tff(fact_8958_ATP_Olambda__776,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_gj(fun(A,nat),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),if(nat,aa(A,bool,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat)))) ) ).

% ATP.lambda_776
tff(fact_8959_ATP_Olambda__777,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_apo(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_777
tff(fact_8960_ATP_Olambda__778,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,nat),fun(A,bool),aTP_Lamp_gz(fun(A,nat),fun(fun(A,nat),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub)))) ) ).

% ATP.lambda_778
tff(fact_8961_ATP_Olambda__779,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kb(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_779
tff(fact_8962_ATP_Olambda__780,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ka(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_780
tff(fact_8963_ATP_Olambda__781,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aoj(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_781
tff(fact_8964_ATP_Olambda__782,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_mb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_782
tff(fact_8965_ATP_Olambda__783,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_eu(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_783
tff(fact_8966_ATP_Olambda__784,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_784
tff(fact_8967_ATP_Olambda__785,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_iw(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_785
tff(fact_8968_ATP_Olambda__786,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_vv(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ).

% ATP.lambda_786
tff(fact_8969_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(multiset(A),B),Uua: fun(C,multiset(A)),Uub: C] : aa(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_aly(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),Uu),Uua),Uub) = aa(multiset(A),B,Uu,aa(C,multiset(A),Uua,Uub)) ) ).

% ATP.lambda_787
tff(fact_8970_ATP_Olambda__788,axiom,
    ! [V4: $tType,U3: $tType,T: $tType,Uu: fun(U3,set(V4)),Uua: fun(T,U3),Uub: T] : aa(T,set(V4),aa(fun(T,U3),fun(T,set(V4)),aTP_Lamp_na(fun(U3,set(V4)),fun(fun(T,U3),fun(T,set(V4))),Uu),Uua),Uub) = aa(U3,set(V4),Uu,aa(T,U3,Uua,Uub)) ).

% ATP.lambda_788
tff(fact_8971_ATP_Olambda__789,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_789
tff(fact_8972_ATP_Olambda__790,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pc(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_790
tff(fact_8973_ATP_Olambda__791,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_be(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).

% ATP.lambda_791
tff(fact_8974_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_cw(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_792
tff(fact_8975_ATP_Olambda__793,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sg(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_793
tff(fact_8976_ATP_Olambda__794,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Uu: fun(A,B),Uua: fun(B,A),Uub: B] : aa(B,B,aa(fun(B,A),fun(B,B),aTP_Lamp_adc(fun(A,B),fun(fun(B,A),fun(B,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_794
tff(fact_8977_ATP_Olambda__795,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aol(fun(A,bool),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uu,aa(B,A,Uua,Uub))) ) ).

% ATP.lambda_795
tff(fact_8978_ATP_Olambda__796,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,bool),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_aks(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),Uu),Uua),Uub) = aa(A,fun(B,bool),Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_796
tff(fact_8979_ATP_Olambda__797,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: D] : aa(D,fun(B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_ww(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),Uu),Uua),Uub) = aa(A,fun(B,C),Uu,aa(D,A,Uua,Uub)) ).

% ATP.lambda_797
tff(fact_8980_ATP_Olambda__798,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(num,A),Uub: num] : aa(num,B,aa(fun(num,A),fun(num,B),aTP_Lamp_kw(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_798
tff(fact_8981_ATP_Olambda__799,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_kv(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).

% ATP.lambda_799
tff(fact_8982_ATP_Olambda__800,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_anh(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_800
tff(fact_8983_ATP_Olambda__801,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ano(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_801
tff(fact_8984_ATP_Olambda__802,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_anp(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_802
tff(fact_8985_ATP_Olambda__803,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_pj(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_803
tff(fact_8986_ATP_Olambda__804,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(B,A),Uub: A] : aa(A,A,aa(fun(B,A),fun(A,A),aTP_Lamp_adb(fun(A,B),fun(fun(B,A),fun(A,A)),Uu),Uua),Uub) = aa(B,A,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_804
tff(fact_8987_ATP_Olambda__805,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_acp(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uua,aa(A,B,Uu,Uub))) ) ).

% ATP.lambda_805
tff(fact_8988_ATP_Olambda__806,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_le(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_806
tff(fact_8989_ATP_Olambda__807,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqg(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ).

% ATP.lambda_807
tff(fact_8990_ATP_Olambda__808,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_yo(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).

% ATP.lambda_808
tff(fact_8991_ATP_Olambda__809,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C,Uub: B] : aa(B,fun(A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_abs(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).

% ATP.lambda_809
tff(fact_8992_ATP_Olambda__810,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_wn(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).

% ATP.lambda_810
tff(fact_8993_ATP_Olambda__811,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_aeo(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_811
tff(fact_8994_ATP_Olambda__812,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,heap_Time_Heap(A)),Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_qo(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(B,heap_Time_Heap(A),Uua,Uu) ).

% ATP.lambda_812
tff(fact_8995_ATP_Olambda__813,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_ajq(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ajp(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).

% ATP.lambda_813
tff(fact_8996_ATP_Olambda__814,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: A] : aa(A,option(A),aa(A,fun(A,option(A)),aTP_Lamp_bv(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Uu,Uua),Uub)) ).

% ATP.lambda_814
tff(fact_8997_ATP_Olambda__815,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(B),Uub: C] : aa(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_apr(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),Uu),Uua),Uub) = comm_m9189036328036947845d_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(C,fun(B,A),aTP_Lamp_ek(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_815
tff(fact_8998_ATP_Olambda__816,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(B),Uub: C] : aa(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_akd(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(C,fun(B,A),aTP_Lamp_ds(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_816
tff(fact_8999_ATP_Olambda__817,axiom,
    ! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: product_prod(B,C)] : aa(product_prod(B,C),set(A),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_ox(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E)),set(set(A)),image2(product_prod(D,E),set(A),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_ow(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),set(A))),Uu),Uub)),Uua)) ).

% ATP.lambda_817
tff(fact_9000_ATP_Olambda__818,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_mu(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(C,fun(B,A),aTP_Lamp_mt(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_818
tff(fact_9001_ATP_Olambda__819,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_nv(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(C,fun(B,A),aTP_Lamp_mt(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_819
tff(fact_9002_ATP_Olambda__820,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(C),Uub: B] : aa(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_apq(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),Uu),Uua),Uub) = comm_m9189036328036947845d_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_820
tff(fact_9003_ATP_Olambda__821,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(C),Uub: B] : aa(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_akc(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_821
tff(fact_9004_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_ms(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_822
tff(fact_9005_ATP_Olambda__823,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,set(C))),Uua: set(B),Uub: A] : aa(A,set(C),aa(set(B),fun(A,set(C)),aTP_Lamp_ala(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),Uu),Uua),Uub) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uu,Uub)),Uua)) ).

% ATP.lambda_823
tff(fact_9006_ATP_Olambda__824,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_nu(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_824
tff(fact_9007_ATP_Olambda__825,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_qu(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_825
tff(fact_9008_ATP_Olambda__826,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_qv(B,fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_826
tff(fact_9009_ATP_Olambda__827,axiom,
    ! [S7: $tType,R7: $tType,Q8: $tType,Uu: fun(R7,set(S7)),Uua: fun(Q8,set(R7)),Uub: Q8] : aa(Q8,set(S7),aa(fun(Q8,set(R7)),fun(Q8,set(S7)),aTP_Lamp_my(fun(R7,set(S7)),fun(fun(Q8,set(R7)),fun(Q8,set(S7))),Uu),Uua),Uub) = aa(set(set(S7)),set(S7),complete_Sup_Sup(set(S7)),aa(set(R7),set(set(S7)),image2(R7,set(S7),Uu),aa(Q8,set(R7),Uua,Uub))) ).

% ATP.lambda_827
tff(fact_9010_ATP_Olambda__828,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_mw(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Uu),aa(C,set(B),Uua,Uub))) ).

% ATP.lambda_828
tff(fact_9011_ATP_Olambda__829,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_nk(fun(B,A),fun(fun(C,set(B)),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,Uu),aa(C,set(B),Uua,Uub))) ) ).

% ATP.lambda_829
tff(fact_9012_ATP_Olambda__830,axiom,
    ! [S7: $tType,R7: $tType,Q8: $tType,Uu: fun(R7,set(S7)),Uua: fun(Q8,set(R7)),Uub: Q8] : aa(Q8,set(S7),aa(fun(Q8,set(R7)),fun(Q8,set(S7)),aTP_Lamp_oj(fun(R7,set(S7)),fun(fun(Q8,set(R7)),fun(Q8,set(S7))),Uu),Uua),Uub) = aa(set(set(S7)),set(S7),complete_Inf_Inf(set(S7)),aa(set(R7),set(set(S7)),image2(R7,set(S7),Uu),aa(Q8,set(R7),Uua,Uub))) ).

% ATP.lambda_830
tff(fact_9013_ATP_Olambda__831,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ip(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_831
tff(fact_9014_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_yq(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),Uu),Uua)) ).

% ATP.lambda_832
tff(fact_9015_ATP_Olambda__833,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(A,fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_rn(fun(A,A),fun(A,fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_833
tff(fact_9016_ATP_Olambda__834,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_anl(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_834
tff(fact_9017_ATP_Olambda__835,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_kz(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ky(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub))) ).

% ATP.lambda_835
tff(fact_9018_ATP_Olambda__836,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
      ( pp(aa(A,bool,aa(set(C),fun(A,bool),aTP_Lamp_zw(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [B13: B] : pp(aa(set(C),bool,member(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B13)),Uua)) ) ).

% ATP.lambda_836
tff(fact_9019_ATP_Olambda__837,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_abi(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [I4: 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),I4)),aa(nat,B,nth(B,Uua),I4)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua)))) ) ) ).

% ATP.lambda_837
tff(fact_9020_ATP_Olambda__838,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_abe(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [A8: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),aa(A,B,Uua,A8)) )
          & pp(aa(set(A),bool,member(A,A8),Uu)) ) ) ).

% ATP.lambda_838
tff(fact_9021_ATP_Olambda__839,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_ack(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I4: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu)))
          & pp(aa(set(nat),bool,member(nat,I4),Uua)) ) ) ).

% ATP.lambda_839
tff(fact_9022_ATP_Olambda__840,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_aao(nat,fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I4: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uua),I4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uu),I4))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uua))) ) ) ).

% ATP.lambda_840
tff(fact_9023_ATP_Olambda__841,axiom,
    ! [A: $tType,Uu: set(set(product_prod(A,A))),Uua: set(product_prod(A,A)),Uub: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_anu(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua),Uub))
    <=> ? [R4: set(product_prod(A,A))] :
          ( ( Uub = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R4),Uua) )
          & pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),R4),Uu)) ) ) ).

% ATP.lambda_841
tff(fact_9024_ATP_Olambda__842,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aei(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A8: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A8) )
              & pp(aa(set(A),bool,member(A,A8),Uu)) ) ) ) ).

% ATP.lambda_842
tff(fact_9025_ATP_Olambda__843,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_zz(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [B13: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B13) )
              & pp(aa(set(A),bool,member(A,B13),Uua)) ) ) ) ).

% ATP.lambda_843
tff(fact_9026_ATP_Olambda__844,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeg(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A8: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A8) )
              & pp(aa(set(A),bool,member(A,A8),Uu)) ) ) ) ).

% ATP.lambda_844
tff(fact_9027_ATP_Olambda__845,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_amp(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A8: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uua),A8) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A8),Uu)) ) ) ).

% ATP.lambda_845
tff(fact_9028_ATP_Olambda__846,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_amu(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A8: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uua),A8) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A8),Uu)) ) ) ).

% ATP.lambda_846
tff(fact_9029_ATP_Olambda__847,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_aaw(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [I4: nat] :
          ( ( Uub = aa(B,set(A),Uu,aa(nat,B,nth(B,Uua),I4)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_847
tff(fact_9030_ATP_Olambda__848,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_abd(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [A8: B] :
          ( ( Uub = aa(B,set(A),Uu,A8) )
          & pp(aa(set(B),bool,member(B,A8),aa(list(B),set(B),set2(B),Uua))) ) ) ).

% ATP.lambda_848
tff(fact_9031_ATP_Olambda__849,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_zj(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [L2: B] :
          ( ( Uub = aa(B,A,Uu,L2) )
          & pp(aa(set(B),bool,member(B,L2),Uua)) ) ) ).

% ATP.lambda_849
tff(fact_9032_ATP_Olambda__850,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: fun(C,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(C,bool),fun(B,bool),aTP_Lamp_aho(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X4: C] :
          ( pp(aa(set(C),bool,member(C,X4),aa(B,set(C),Uu,Uub)))
         => pp(aa(C,bool,Uua,X4)) ) ) ).

% ATP.lambda_850
tff(fact_9033_ATP_Olambda__851,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,bool),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(fun(B,bool),fun(set(A),bool),aTP_Lamp_zl(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( ( Uub = aa(B,set(A),Uu,X4) )
          & pp(aa(B,bool,Uua,X4)) ) ) ).

% ATP.lambda_851
tff(fact_9034_ATP_Olambda__852,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_zx(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( ( Uub = aa(B,A,Uu,X4) )
          & pp(aa(B,bool,Uua,X4)) ) ) ).

% ATP.lambda_852
tff(fact_9035_ATP_Olambda__853,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,B),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aTP_Lamp_zs(fun(A,bool),fun(fun(A,B),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( ( Uub = aa(A,B,Uua,X4) )
          & pp(aa(A,bool,Uu,X4)) ) ) ).

% ATP.lambda_853
tff(fact_9036_ATP_Olambda__854,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,fun(A,bool)),Uub: C] :
      ( pp(aa(C,bool,aa(fun(C,fun(A,bool)),fun(C,bool),aTP_Lamp_aog(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(C,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))))
          & pp(aa(A,bool,aa(C,fun(A,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_854
tff(fact_9037_ATP_Olambda__855,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,C),Uub: fun(A,C)] :
      ( pp(aa(fun(A,C),bool,aa(fun(A,C),fun(fun(A,C),bool),aTP_Lamp_aoi(fun(A,fun(B,bool)),fun(fun(A,C),fun(fun(A,C),bool)),Uu),Uua),Uub))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))))
         => ( aa(A,C,Uua,X4) = aa(A,C,Uub,X4) ) ) ) ).

% ATP.lambda_855
tff(fact_9038_ATP_Olambda__856,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
      ( pp(aa(A,bool,aa(list(B),fun(A,bool),aTP_Lamp_aak(fun(B,option(A)),fun(list(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),aa(list(B),set(B),set2(B),Uua)))
          & ( aa(B,option(A),Uu,X4) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_856
tff(fact_9039_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_ahm(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uu))
         => pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_857
tff(fact_9040_ATP_Olambda__858,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aow(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uu))
         => pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_858
tff(fact_9041_ATP_Olambda__859,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_aim(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uu))
          & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_859
tff(fact_9042_ATP_Olambda__860,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aii(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uu))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_860
tff(fact_9043_ATP_Olambda__861,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_aij(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uua))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uu,X4),Uub)) ) ) ).

% ATP.lambda_861
tff(fact_9044_ATP_Olambda__862,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_alf(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),Uu))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ale(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)),X4)) ) ) ).

% ATP.lambda_862
tff(fact_9045_ATP_Olambda__863,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_zv(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [P4: product_prod(A,B)] :
          ( ( Uub = P4 )
          & pp(aa(set(C),bool,member(C,aa(B,C,aa(A,fun(B,C),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_863
tff(fact_9046_ATP_Olambda__864,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_aiq(set(product_prod(A,B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Uub)),Uu)) ) ) ).

% ATP.lambda_864
tff(fact_9047_ATP_Olambda__865,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ail(fun(A,B),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & ( aa(A,B,Uu,X4) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_865
tff(fact_9048_ATP_Olambda__866,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_aiv(fun(A,option(B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & ( aa(A,option(B),Uu,X4) = aa(B,option(B),some(B),Uub) ) ) ) ).

% ATP.lambda_866
tff(fact_9049_ATP_Olambda__867,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_ahl(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uua))
         => pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X4))) ) ) ).

% ATP.lambda_867
tff(fact_9050_ATP_Olambda__868,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_ain(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uua))
          & pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X4))) ) ) ).

% ATP.lambda_868
tff(fact_9051_ATP_Olambda__869,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_aio(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & ( Uub = aa(A,B,Uu,X4) ) ) ) ).

% ATP.lambda_869
tff(fact_9052_ATP_Olambda__870,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: filter(B),Uub: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(filter(B),fun(fun(A,bool),bool),aTP_Lamp_aqe(fun(A,B),fun(filter(B),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ? [Q9: fun(B,bool)] :
          ( eventually(B,Q9,Uua)
          & ! [X4: A] :
              ( pp(aa(B,bool,Q9,aa(A,B,Uu,X4)))
             => pp(aa(A,bool,Uub,X4)) ) ) ) ).

% ATP.lambda_870
tff(fact_9053_ATP_Olambda__871,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_zg(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [Y5: A] :
          ( pp(aa(A,bool,Uu,Y5))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),Y5)) ) ) ).

% ATP.lambda_871
tff(fact_9054_ATP_Olambda__872,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_zm(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(B,bool,Uua,X4))
          & pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X4))) ) ) ).

% ATP.lambda_872
tff(fact_9055_ATP_Olambda__873,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afb(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),Uu),Uua) ) ).

% ATP.lambda_873
tff(fact_9056_ATP_Olambda__874,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,bool)),Uub: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_zk(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A8: B,B13: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A8)),aa(B,A,Uu,B13)) )
          & pp(aa(B,bool,aa(B,fun(B,bool),Uua,A8),B13)) ) ) ).

% ATP.lambda_874
tff(fact_9057_ATP_Olambda__875,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_agz(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A19: B,A25: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A19)),aa(B,A,Uua,A25)) )
          & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A19),A25)),Uu)) ) ) ).

% ATP.lambda_875
tff(fact_9058_ATP_Olambda__876,axiom,
    ! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: fun(A,A2),Uub: product_prod(A2,A2)] :
      ( pp(aa(product_prod(A2,A2),bool,aa(fun(A,A2),fun(product_prod(A2,A2),bool),aTP_Lamp_ark(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),Uu),Uua),Uub))
    <=> ? [A8: A,B13: A] :
          ( ( Uub = aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,Uua,A8)),aa(A,A2,Uua,B13)) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A8),B13)),Uu)) ) ) ).

% ATP.lambda_876
tff(fact_9059_ATP_Olambda__877,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_aav(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [A8: A,V5: 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),A8),V5)) )
          | ? [U7: list(A),Aa3: A,B13: A,Va3: list(A),W3: list(A)] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B13)),Uu))
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U7),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa3),Va3)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U7),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B13),W3)) ) ) ) ) ).

% ATP.lambda_877
tff(fact_9060_ATP_Olambda__878,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aeh(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A8: A,B13: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A8),B13) )
              & pp(aa(set(A),bool,member(A,A8),Uu))
              & pp(aa(set(A),bool,member(A,B13),Uua)) ) ) ) ).

% ATP.lambda_878
tff(fact_9061_ATP_Olambda__879,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aef(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A8: A,B13: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A8),B13) )
              & pp(aa(set(A),bool,member(A,A8),Uu))
              & pp(aa(set(A),bool,member(A,B13),Uua)) ) ) ) ).

% ATP.lambda_879
tff(fact_9062_ATP_Olambda__880,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_amo(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A8: multiset(A),B13: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A8),B13) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A8),Uu))
          & pp(aa(set(multiset(A)),bool,member(multiset(A),B13),Uua)) ) ) ).

% ATP.lambda_880
tff(fact_9063_ATP_Olambda__881,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_amv(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A8: multiset(A),B13: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A8),B13) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A8),Uu))
          & pp(aa(set(multiset(A)),bool,member(multiset(A),B13),Uua)) ) ) ).

% ATP.lambda_881
tff(fact_9064_ATP_Olambda__882,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(list(A)),fun(list(A),bool),aTP_Lamp_zp(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [X4: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
          & pp(aa(set(A),bool,member(A,X4),Uu))
          & pp(aa(set(list(A)),bool,member(list(A),Xs3),Uua)) ) ) ).

% ATP.lambda_882
tff(fact_9065_ATP_Olambda__883,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_apn(filter(A),fun(filter(A),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ? [Q9: fun(A,bool),R10: fun(A,bool)] :
          ( eventually(A,Q9,Uu)
          & eventually(A,R10,Uua)
          & ! [X4: A] :
              ( ( pp(aa(A,bool,Q9,X4))
                & pp(aa(A,bool,R10,X4)) )
             => pp(aa(A,bool,Uub,X4)) ) ) ) ).

% ATP.lambda_883
tff(fact_9066_ATP_Olambda__884,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: multiset(A),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_ali(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A8: A,M03: multiset(A),K9: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A8),M03) )
          & ( Uua = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M03),K9) )
          & ! [B13: A] :
              ( pp(aa(set(A),bool,member(A,B13),aa(multiset(A),set(A),set_mset(A),K9)))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),A8)),Uu)) ) ) ) ).

% ATP.lambda_884
tff(fact_9067_ATP_Olambda__885,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_aac(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [Us3: list(A),Z6: A,Z11: A,Vs3: list(A)] :
          ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z6),Vs3)) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),Z11)),Uu))
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z11),Vs3)) ) ) ) ).

% ATP.lambda_885
tff(fact_9068_ATP_Olambda__886,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_iu(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_886
tff(fact_9069_ATP_Olambda__887,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_xe(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(product_prod(C,A),A,product_snd(C,A),Uu)),Uua),aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_887
tff(fact_9070_ATP_Olambda__888,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_agj(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_888
tff(fact_9071_ATP_Olambda__889,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_agi(set(A),fun(fun(A,B),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uua),Uu)),the_inv_into(A,B,Uu,Uua,Uuc),Uub) ).

% ATP.lambda_889
tff(fact_9072_ATP_Olambda__890,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_xl(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = if(set(B),aa(set(A),bool,member(A,Uua),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uub),Uuc),Uuc) ).

% ATP.lambda_890
tff(fact_9073_ATP_Olambda__891,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_ff(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_891
tff(fact_9074_ATP_Olambda__892,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_ew(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_892
tff(fact_9075_ATP_Olambda__893,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_ex(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_893
tff(fact_9076_ATP_Olambda__894,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_fg(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_894
tff(fact_9077_ATP_Olambda__895,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_aex(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,member(A,Uuc),Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_895
tff(fact_9078_ATP_Olambda__896,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qk(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_896
tff(fact_9079_ATP_Olambda__897,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qj(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_897
tff(fact_9080_ATP_Olambda__898,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_lh(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).

% ATP.lambda_898
tff(fact_9081_ATP_Olambda__899,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_afj(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uu),aa(B,option(B),some(B),Uua),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_899
tff(fact_9082_ATP_Olambda__900,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B,Uuc: A] : aa(A,option(B),aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_qa(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uua),aa(B,option(B),some(B),Uub),aa(A,option(B),Uu,Uuc)) ).

% ATP.lambda_900
tff(fact_9083_ATP_Olambda__901,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: B,Uub: set(A),Uuc: A] : aa(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_acq(B,fun(B,fun(set(A),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,member(A,Uuc),Uub),Uu,Uua) ).

% ATP.lambda_901
tff(fact_9084_ATP_Olambda__902,axiom,
    ! [A: $tType,Uu: fun(A,bool),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_wf(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = if(product_prod(list(A),list(A)),aa(A,bool,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_902
tff(fact_9085_ATP_Olambda__903,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_hs(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_903
tff(fact_9086_ATP_Olambda__904,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_hr(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_904
tff(fact_9087_ATP_Olambda__905,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ct(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_905
tff(fact_9088_ATP_Olambda__906,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,option(B)),Uub: fun(A,option(B)),Uuc: A] : aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_afm(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,Uu,Uuc),aa(A,option(B),Uua,Uuc),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_906
tff(fact_9089_ATP_Olambda__907,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_ani(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_907
tff(fact_9090_ATP_Olambda__908,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_xh(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_xg(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_908
tff(fact_9091_ATP_Olambda__909,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_xd(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_xc(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_909
tff(fact_9092_ATP_Olambda__910,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_po(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uuc),aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_910
tff(fact_9093_ATP_Olambda__911,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,fun(list(B),fun(C,C))),Uua: fun(A,B),Uub: A,Uuc: list(A)] : aa(list(A),fun(C,C),aa(A,fun(list(A),fun(C,C)),aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_anm(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Uu),Uua),Uub),Uuc) = aa(list(B),fun(C,C),aa(B,fun(list(B),fun(C,C)),Uu,aa(A,B,Uua,Uub)),aa(list(A),list(B),map(A,B,Uua),Uuc)) ).

% ATP.lambda_911
tff(fact_9094_ATP_Olambda__912,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_uo(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ).

% ATP.lambda_912
tff(fact_9095_ATP_Olambda__913,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_abn(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).

% ATP.lambda_913
tff(fact_9096_ATP_Olambda__914,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aau(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_914
tff(fact_9097_ATP_Olambda__915,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahi(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).

% ATP.lambda_915
tff(fact_9098_ATP_Olambda__916,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,B),Uub: A,Uuc: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_akr(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(C,B,Uua,Uuc))) ) ).

% ATP.lambda_916
tff(fact_9099_ATP_Olambda__917,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,A)),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,A,aa(A,fun(C,A),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_un(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu,Uub),aa(C,B,Uua,Uuc)) ).

% ATP.lambda_917
tff(fact_9100_ATP_Olambda__918,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_dw(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_dv(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_918
tff(fact_9101_ATP_Olambda__919,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: filter(A),Uuc: fun(B,bool)] :
      ( pp(aa(fun(B,bool),bool,aa(filter(A),fun(fun(B,bool),bool),aa(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),aTP_Lamp_apm(set(A),fun(fun(A,B),fun(filter(A),fun(fun(B,bool),bool))),Uu),Uua),Uub),Uuc))
    <=> eventually(A,aa(fun(B,bool),fun(A,bool),aa(fun(A,B),fun(fun(B,bool),fun(A,bool)),aTP_Lamp_apl(set(A),fun(fun(A,B),fun(fun(B,bool),fun(A,bool))),Uu),Uua),Uuc),Uub) ) ).

% ATP.lambda_919
tff(fact_9102_ATP_Olambda__920,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B,Uuc: B] : aa(B,fun(list(B),list(B)),aa(B,fun(B,fun(list(B),list(B))),aa(list(B),fun(B,fun(B,fun(list(B),list(B)))),aTP_Lamp_tp(fun(B,A),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub),Uuc) = case_list(list(B),B,if(list(B),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uu,Uuc)),Uua,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uuc),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),nil(B)))),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_to(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua)) ) ).

% ATP.lambda_920
tff(fact_9103_ATP_Olambda__921,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: bool,Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa(bool,fun(A,fun(B,assn)),aTP_Lamp_ba(fun(A,fun(B,assn)),fun(bool,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_921
tff(fact_9104_ATP_Olambda__922,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(D,E),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(product_prod(D,E),fun(B,fun(C,set(A))),aTP_Lamp_ov(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(D,E),set(A),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc)),Uua) ).

% ATP.lambda_922
tff(fact_9105_ATP_Olambda__923,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_fz(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(C,fun(B,A),aTP_Lamp_ek(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc)),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_fw(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_923
tff(fact_9106_ATP_Olambda__924,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_fx(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(C,fun(B,A),aTP_Lamp_ds(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc)),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_fw(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_924
tff(fact_9107_ATP_Olambda__925,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_apw(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uu),Uua),Uub),Uuc) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_Pair(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),Uu)),Uuc)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Uub),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),Uua))) ).

% ATP.lambda_925
tff(fact_9108_ATP_Olambda__926,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: set(A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_alh(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(product_prod(A,A)),bool,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_yj(set(A),fun(A,set(A)),Uua))))
        & pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uuc)) ) ) ).

% ATP.lambda_926
tff(fact_9109_ATP_Olambda__927,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_qz(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_927
tff(fact_9110_ATP_Olambda__928,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_abm(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uu))
        & ~ pp(aa(set(A),bool,member(A,Uub),Uua))
        & ~ pp(aa(set(A),bool,member(A,Uuc),Uua)) ) ) ).

% ATP.lambda_928
tff(fact_9111_ATP_Olambda__929,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_js(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
          & pp(aa(set(product_prod(A,A)),bool,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_929
tff(fact_9112_ATP_Olambda__930,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ajw(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(set(product_prod(B,B)),bool,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_930
tff(fact_9113_ATP_Olambda__931,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),bool),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_jr(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
        & pp(aa(product_prod(A,B),bool,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc))) ) ) ).

% ATP.lambda_931
tff(fact_9114_ATP_Olambda__932,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_sx(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uuc)),Uua)),Uub) ) ).

% ATP.lambda_932
tff(fact_9115_ATP_Olambda__933,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A),Uuc: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_aqm(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),Uua,Uuc)),Uub)),Uu) ).

% ATP.lambda_933
tff(fact_9116_ATP_Olambda__934,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_hx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_934
tff(fact_9117_ATP_Olambda__935,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hu(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_935
tff(fact_9118_ATP_Olambda__936,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_hw(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_936
tff(fact_9119_ATP_Olambda__937,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_aas(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X4: A,Y5: A,Xs6: 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),X4),Xs6)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),Uu)) ) ) ) ).

% ATP.lambda_937
tff(fact_9120_ATP_Olambda__938,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_ki(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_938
tff(fact_9121_ATP_Olambda__939,axiom,
    ! [A: $tType,Uu: bool,Uua: A,Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_afh(bool,fun(A,fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( pp(Uu)
         => ( Uuc = Uua ) )
        & ( ~ pp(Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_939
tff(fact_9122_ATP_Olambda__940,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_cx(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),aa(set(B),set(A),image2(B,A,Uu),Uua)))
        & pp(aa(A,bool,Uub,Uuc)) ) ) ).

% ATP.lambda_940
tff(fact_9123_ATP_Olambda__941,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: multiset(B),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(multiset(B),fun(A,fun(B,bool)),aTP_Lamp_aln(fun(B,A),fun(multiset(B),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),aa(multiset(B),set(B),set_mset(B),Uua)))
        & ( Uub = aa(B,A,Uu,Uuc) ) ) ) ).

% ATP.lambda_941
tff(fact_9124_ATP_Olambda__942,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,fun(C,bool)),Uub: B,Uuc: C] :
      ( pp(aa(C,bool,aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_fu(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(C),bool,member(C,Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_942
tff(fact_9125_ATP_Olambda__943,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ky(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_943
tff(fact_9126_ATP_Olambda__944,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,fun(C,bool)),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_fw(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_944
tff(fact_9127_ATP_Olambda__945,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,bool)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ig(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_945
tff(fact_9128_ATP_Olambda__946,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ih(set(A),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_946
tff(fact_9129_ATP_Olambda__947,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_gk(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
        & ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_947
tff(fact_9130_ATP_Olambda__948,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lf(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_948
tff(fact_9131_ATP_Olambda__949,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_afc(fun(A,B),fun(set(A),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uua))
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_949
tff(fact_9132_ATP_Olambda__950,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( pp(aa(B,bool,aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_aep(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uub))
        & pp(aa(set(A),bool,member(A,aa(B,A,Uua,Uuc)),Uu)) ) ) ).

% ATP.lambda_950
tff(fact_9133_ATP_Olambda__951,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_kh(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_951
tff(fact_9134_ATP_Olambda__952,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hh(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_952
tff(fact_9135_ATP_Olambda__953,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: list(B),Uuc: list(B)] : aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(list(B),fun(list(B),fun(list(B),list(B))),aTP_Lamp_tm(fun(B,A),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua),Uub),Uuc) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_sort_key(B,A,Uu),Uua)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uub),aa(list(B),list(B),linorder_sort_key(B,A,Uu),Uuc))) ) ).

% ATP.lambda_953
tff(fact_9136_ATP_Olambda__954,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_afz(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_954
tff(fact_9137_ATP_Olambda__955,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_rf(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_955
tff(fact_9138_ATP_Olambda__956,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_afx(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_956
tff(fact_9139_ATP_Olambda__957,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_rd(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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_957
tff(fact_9140_ATP_Olambda__958,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_rh(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_958
tff(fact_9141_ATP_Olambda__959,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_rj(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_959
tff(fact_9142_ATP_Olambda__960,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_aby(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_960
tff(fact_9143_ATP_Olambda__961,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_abx(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_961
tff(fact_9144_ATP_Olambda__962,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_afe(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_962
tff(fact_9145_ATP_Olambda__963,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_aie(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua) ) ) ).

% ATP.lambda_963
tff(fact_9146_ATP_Olambda__964,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_cy(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uua))
        & pp(aa(A,bool,Uub,aa(B,A,Uu,Uuc))) ) ) ).

% ATP.lambda_964
tff(fact_9147_ATP_Olambda__965,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),aTP_Lamp_mq(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu,Uua))) ) ).

% ATP.lambda_965
tff(fact_9148_ATP_Olambda__966,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_gf(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).

% ATP.lambda_966
tff(fact_9149_ATP_Olambda__967,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_gd(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_967
tff(fact_9150_ATP_Olambda__968,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_km(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_968
tff(fact_9151_ATP_Olambda__969,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,bool),Uub: fun(A,B),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),aTP_Lamp_aph(set(A),fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(B,bool,Uua,aa(A,B,Uub,Uuc)))
        & pp(aa(set(A),bool,member(A,Uuc),Uu)) ) ) ).

% ATP.lambda_969
tff(fact_9152_ATP_Olambda__970,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: fun(B,bool),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aa(fun(A,B),fun(fun(B,bool),fun(A,bool)),aTP_Lamp_apl(set(A),fun(fun(A,B),fun(fun(B,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(B,bool,Uub,aa(A,B,Uua,Uuc)))
        & pp(aa(set(A),bool,member(A,Uuc),Uu)) ) ) ).

% ATP.lambda_970
tff(fact_9153_ATP_Olambda__971,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ya(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_971
tff(fact_9154_ATP_Olambda__972,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_yp(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_yo(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_972
tff(fact_9155_ATP_Olambda__973,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_ajr(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uuc))
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_973
tff(fact_9156_ATP_Olambda__974,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_adk(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_974
tff(fact_9157_ATP_Olambda__975,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( semiring_0(B)
        & comm_monoid_add(A)
        & times(A) )
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ake(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_975
tff(fact_9158_ATP_Olambda__976,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_agx(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).

% ATP.lambda_976
tff(fact_9159_ATP_Olambda__977,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_dv(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_977
tff(fact_9160_ATP_Olambda__978,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_oe(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_978
tff(fact_9161_ATP_Olambda__979,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_nw(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_979
tff(fact_9162_ATP_Olambda__980,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_ng(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_980
tff(fact_9163_ATP_Olambda__981,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_os(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_981
tff(fact_9164_ATP_Olambda__982,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_yk(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_982
tff(fact_9165_ATP_Olambda__983,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_aba(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).

% ATP.lambda_983
tff(fact_9166_ATP_Olambda__984,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_xq(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(B,bool,Uua,Uuc)) ) ) ).

% ATP.lambda_984
tff(fact_9167_ATP_Olambda__985,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_uu(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_985
tff(fact_9168_ATP_Olambda__986,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_fy(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uua,Uuc)),aa(fun(C,bool),set(C),collect(C),aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_fu(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_986
tff(fact_9169_ATP_Olambda__987,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_fv(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uua,Uuc)),aa(fun(C,bool),set(C),collect(C),aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_fu(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_987
tff(fact_9170_ATP_Olambda__988,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B,Uuc: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_to(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(B),product_prod(list(B),list(B))),list(B),aa(fun(list(B),fun(product_prod(list(B),list(B)),list(B))),fun(product_prod(list(B),product_prod(list(B),list(B))),list(B)),product_case_prod(list(B),product_prod(list(B),list(B)),list(B)),aTP_Lamp_tn(fun(B,A),fun(list(B),fun(product_prod(list(B),list(B)),list(B))),Uu)),linorder_part(B,A,Uu,aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Uua)) ) ).

% ATP.lambda_988
tff(fact_9171_ATP_Olambda__989,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_ans(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
       => pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ) ).

% ATP.lambda_989
tff(fact_9172_ATP_Olambda__990,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_apg(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_990
tff(fact_9173_ATP_Olambda__991,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_lg(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lf(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_991
tff(fact_9174_ATP_Olambda__992,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,fun(A,bool)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_zo(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A8: A] :
            ( ( Uub = A8 )
            & ( Uuc = A8 ) )
        | ? [A8: A,B13: A,C5: A] :
            ( ( Uub = A8 )
            & ( Uuc = C5 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A8),B13))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B13),C5)) ) ) ) ).

% ATP.lambda_992
tff(fact_9175_ATP_Olambda__993,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,fun(A,bool)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_zn(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A8: A,B13: A] :
            ( ( Uub = A8 )
            & ( Uuc = B13 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,A8),B13)) )
        | ? [A8: A,B13: A,C5: A] :
            ( ( Uub = A8 )
            & ( Uuc = C5 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A8),B13))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B13),C5)) ) ) ) ).

% ATP.lambda_993
tff(fact_9176_ATP_Olambda__994,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(list(A),fun(list(A),bool)),Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_zq(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [Y5: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) ) )
        | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y5)) )
        | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y5))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,Y5),X4))
            & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uua,Xs3),Ys4)) ) ) ) ).

% ATP.lambda_994
tff(fact_9177_ATP_Olambda__995,axiom,
    ! [A: $tType,B: $tType,Uu: bool,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_az(bool,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_995
tff(fact_9178_ATP_Olambda__996,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_ta(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_996
tff(fact_9179_ATP_Olambda__997,axiom,
    ! [A: $tType,B: $tType,Uu: bool,Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_jw(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(Uu)
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_997
tff(fact_9180_ATP_Olambda__998,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_acn(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_998
tff(fact_9181_ATP_Olambda__999,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B,Uub: heap_ext(product_unit),Uuc: nat] : aa(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_jk(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_timeFrame(A,Uuc,aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,aa(B,heap_Time_Heap(A),Uu,Uua)),Uub)) ).

% ATP.lambda_999
tff(fact_9182_ATP_Olambda__1000,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: A,Uub: heap_ext(product_unit),Uuc: nat] : aa(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_jn(fun(A,heap_Time_Heap(B)),fun(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_timeFrame(B,Uuc,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),Uu,Uua)),Uub)) ).

% ATP.lambda_1000
tff(fact_9183_ATP_Olambda__1001,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(bool),aa(A,fun(list(A),option(bool)),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_ue(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),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_1001
tff(fact_9184_ATP_Olambda__1002,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_gn(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Uua),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_gk(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1002
tff(fact_9185_ATP_Olambda__1003,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_ha(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Uub),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_gk(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1003
tff(fact_9186_ATP_Olambda__1004,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_gl(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),Uua),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_gk(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1004
tff(fact_9187_ATP_Olambda__1005,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_gx(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),Uub),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_gk(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1005
tff(fact_9188_ATP_Olambda__1006,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_abh(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_1006
tff(fact_9189_ATP_Olambda__1007,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(C,D),Uua: fun(A,fun(B,C)),Uub: A,Uuc: B] : aa(B,D,aa(A,fun(B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_iv(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),Uu),Uua),Uub),Uuc) = aa(C,D,Uu,aa(B,C,aa(A,fun(B,C),Uua,Uub),Uuc)) ).

% ATP.lambda_1007
tff(fact_9190_ATP_Olambda__1008,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,fun(list(A),B)),Uub: A,Uuc: list(A)] : aa(list(A),C,aa(A,fun(list(A),C),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_ts(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(list(A),B,aa(A,fun(list(A),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1008
tff(fact_9191_ATP_Olambda__1009,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_1009
tff(fact_9192_ATP_Olambda__1010,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hi(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_1010
tff(fact_9193_ATP_Olambda__1011,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1011
tff(fact_9194_ATP_Olambda__1012,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_df(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1012
tff(fact_9195_ATP_Olambda__1013,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: nat,Uub: list(A),Uuc: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_vu(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,take(A,Uua,Uub)),Uuc))) ) ).

% ATP.lambda_1013
tff(fact_9196_ATP_Olambda__1014,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_abq(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_1014
tff(fact_9197_ATP_Olambda__1015,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_abr(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D,C,Uua,Uuc))) ).

% ATP.lambda_1015
tff(fact_9198_ATP_Olambda__1016,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_tv(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),Uu),Uua),Uub),Uuc) = aa(fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_case_prod(list(B),list(B),product_prod(list(B),product_prod(list(B),list(B)))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_tu(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc)) ) ).

% ATP.lambda_1016
tff(fact_9199_ATP_Olambda__1017,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_rt(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_rs(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1017
tff(fact_9200_ATP_Olambda__1018,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_rq(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_rp(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1018
tff(fact_9201_ATP_Olambda__1019,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( semiring_0(B)
        & comm_monoid_add(A)
        & times(A) )
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: multiset(C),Uuc: A] : aa(A,B,aa(multiset(C),fun(A,B),aa(fun(C,B),fun(multiset(C),fun(A,B)),aTP_Lamp_akf(fun(A,B),fun(fun(C,B),fun(multiset(C),fun(A,B))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ake(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1019
tff(fact_9202_ATP_Olambda__1020,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ot(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_os(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1020
tff(fact_9203_ATP_Olambda__1021,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_nh(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_ng(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1021
tff(fact_9204_ATP_Olambda__1022,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_of(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_oe(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1022
tff(fact_9205_ATP_Olambda__1023,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_nx(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_nw(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1023
tff(fact_9206_ATP_Olambda__1024,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_age(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_1024
tff(fact_9207_ATP_Olambda__1025,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_agc(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_1025
tff(fact_9208_ATP_Olambda__1026,axiom,
    ! [A: $tType,Uu: fun(A,bool),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_vz(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,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_1026
tff(fact_9209_ATP_Olambda__1027,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_uc(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_1027
tff(fact_9210_ATP_Olambda__1028,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_aft(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_1028
tff(fact_9211_ATP_Olambda__1029,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_afv(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_1029
tff(fact_9212_ATP_Olambda__1030,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(C))),Uub: B,Uuc: A] : aa(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_akx(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uua,Uuc)),aa(set(B),set(B),image(B,B,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uub),bot_bot(set(B)))))) ).

% ATP.lambda_1030
tff(fact_9213_ATP_Olambda__1031,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(set(product_prod(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_ou(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E)),set(set(A)),image2(product_prod(D,E),set(A),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc))),Uua)) ).

% ATP.lambda_1031
tff(fact_9214_ATP_Olambda__1032,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_aix(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X4: A] :
          ( ( pp(aa(set(A),bool,member(A,X4),Uu))
           => pp(aa(set(B),bool,member(B,aa(A,B,Uuc,X4)),Uua)) )
          & ( ~ pp(aa(set(A),bool,member(A,X4),Uu))
           => ( aa(A,B,Uuc,X4) = Uub ) ) ) ) ).

% ATP.lambda_1032
tff(fact_9215_ATP_Olambda__1033,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_aai(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [Y5: C] :
          ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y5)),Uu))
          & pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y5),Uuc)),Uua)) ) ) ).

% ATP.lambda_1033
tff(fact_9216_ATP_Olambda__1034,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_aat(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A8: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A8)),aa(C,B,Uub,A8)) )
          & pp(aa(set(C),bool,member(C,A8),Uu)) ) ) ).

% ATP.lambda_1034
tff(fact_9217_ATP_Olambda__1035,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_zi(fun(B,A),fun(B,fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [L2: B] :
          ( ( Uuc = aa(B,A,Uu,L2) )
          & ( ( L2 = Uua )
            | pp(aa(set(B),bool,member(B,L2),Uub)) ) ) ) ).

% ATP.lambda_1035
tff(fact_9218_ATP_Olambda__1036,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A,Uub: fun(A,B),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_aad(A,fun(A,fun(fun(A,B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ? [I4: A] :
              ( ( Uuc = aa(A,B,Uub,I4) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),I4))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I4),Uua)) ) ) ) ).

% ATP.lambda_1036
tff(fact_9219_ATP_Olambda__1037,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,C)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(B,C)),fun(A,fun(A,bool)),aTP_Lamp_akz(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X4: product_prod(B,B)] :
          ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),X4),Uu))
         => pp(aa(product_prod(B,B),bool,aa(fun(B,fun(B,bool)),fun(product_prod(B,B),bool),product_case_prod(B,B,bool),aa(A,fun(B,fun(B,bool)),aa(A,fun(A,fun(B,fun(B,bool))),aTP_Lamp_aky(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,bool)))),Uua),Uub),Uuc)),X4)) ) ) ).

% ATP.lambda_1037
tff(fact_9220_ATP_Olambda__1038,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( pp(aa(C,bool,aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_zt(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [X4: A,Y5: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X4),Y5) )
          & pp(aa(A,bool,Uu,X4))
          & pp(aa(B,bool,Uua,Y5)) ) ) ).

% ATP.lambda_1038
tff(fact_9221_ATP_Olambda__1039,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_agy(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A19: A,A25: A] :
          ( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A19),A25) )
          & pp(aa(set(A),bool,member(A,A19),Uu))
          & pp(aa(set(A),bool,member(A,A25),Uu))
          & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A19)),aa(A,B,Uub,A25))),Uua)) ) ) ).

% ATP.lambda_1039
tff(fact_9222_ATP_Olambda__1040,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_xg(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uua),Uub),aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1040
tff(fact_9223_ATP_Olambda__1041,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_xc(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(A,C)),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uua),Uub),aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1041
tff(fact_9224_ATP_Olambda__1042,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),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_wg(fun(A,fun(A,bool)),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_1042
tff(fact_9225_ATP_Olambda__1043,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,Uu: fun(D,fun(E,C)),Uua: fun(A,D),Uub: fun(B,E),Uuc: A,Uud: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_abb(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(E,C,aa(D,fun(E,C),Uu,aa(A,D,Uua,Uuc)),aa(B,E,Uub,Uud)) ).

% ATP.lambda_1043
tff(fact_9226_ATP_Olambda__1044,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(E,C),Uuc: D,Uud: E] : aa(E,A,aa(D,fun(E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aaz(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(E,C,Uub,Uud)) ).

% ATP.lambda_1044
tff(fact_9227_ATP_Olambda__1045,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(C,A)),Uua: fun(A,fun(B,fun(C,B))),Uub: A,Uuc: B,Uud: C] : aa(C,product_prod(A,B),aa(B,fun(C,product_prod(A,B)),aa(A,fun(B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_wp(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,aa(A,fun(C,A),Uu,Uub),Uud)),aa(C,B,aa(B,fun(C,B),aa(A,fun(B,fun(C,B)),Uua,Uub),Uuc),Uud)) ).

% ATP.lambda_1045
tff(fact_9228_ATP_Olambda__1046,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,C)),Uua: A,Uub: A,Uuc: B,Uud: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(A,fun(B,fun(B,bool)),aa(A,fun(A,fun(B,fun(B,bool))),aTP_Lamp_aky(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( aa(B,C,aa(A,fun(B,C),Uu,Uua),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uud) ) ) ).

% ATP.lambda_1046
tff(fact_9229_ATP_Olambda__1047,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_ed(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_1047
tff(fact_9230_ATP_Olambda__1048,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_ec(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_1048
tff(fact_9231_ATP_Olambda__1049,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_zd(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_trancl(A,Uub)))
          | ( Uuc = Uu ) )
        & ( pp(aa(set(product_prod(A,A)),bool,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_1049
tff(fact_9232_ATP_Olambda__1050,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ef(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,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_1050
tff(fact_9233_ATP_Olambda__1051,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ajp(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uud),bot_bot(set(A))))))),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = Uuc )
            & pp(aa(set(product_prod(A,A)),bool,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_1051
tff(fact_9234_ATP_Olambda__1052,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aan(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_rtrancl(A,Uub)))
        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_rtrancl(A,Uub))) ) ) ).

% ATP.lambda_1052
tff(fact_9235_ATP_Olambda__1053,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ale(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( ( Uub = Uuc )
       => pp(aa(set(product_prod(A,A)),bool,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_1053
tff(fact_9236_ATP_Olambda__1054,axiom,
    ! [C: $tType,B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,fun(A,bool)),Uub: fun(A,fun(E,bool)),Uuc: C,Uud: E] :
      ( pp(aa(E,bool,aa(C,fun(E,bool),aa(fun(A,fun(E,bool)),fun(C,fun(E,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool))),aTP_Lamp_aqt(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))))
          & pp(aa(A,bool,aa(C,fun(A,bool),Uua,Uuc),X4))
          & pp(aa(E,bool,aa(A,fun(E,bool),Uub,X4),Uud)) ) ) ).

% ATP.lambda_1054
tff(fact_9237_ATP_Olambda__1055,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
          ( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_afa(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ? [A8: A,B13: C] :
              ( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A8)),aa(C,B,Uua,B13)) )
              & pp(aa(set(A),bool,member(A,A8),Uub))
              & pp(aa(set(C),bool,member(C,B13),Uuc)) ) ) ) ).

% ATP.lambda_1055
tff(fact_9238_ATP_Olambda__1056,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B),Uud: list(B),Uue: list(B)] : aa(list(B),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_tu(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uuc)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),Uue)),if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub)),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uue))),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uud)),Uue)))) ) ).

% ATP.lambda_1056
tff(fact_9239_ATP_Olambda__1057,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_rp(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),Uu))
        | ( ( Uub = Uud )
          & pp(aa(set(product_prod(B,B)),bool,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_1057
tff(fact_9240_ATP_Olambda__1058,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_rs(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( ( Uub = Uud )
        & pp(aa(A,bool,Uu,Uud))
        & pp(aa(set(product_prod(B,B)),bool,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_1058
tff(fact_9241_ATP_Olambda__1059,axiom,
    ! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_pq(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(Uu) ) ).

% ATP.lambda_1059
tff(fact_9242_ATP_Olambda__1060,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_qn(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).

% ATP.lambda_1060
tff(fact_9243_ATP_Olambda__1061,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(B)
     => ! [Uu: heap_Time_Heap(A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_qs(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1061
tff(fact_9244_ATP_Olambda__1062,axiom,
    ! [B: $tType,A: $tType,Uu: multiset(A),Uua: B] : aa(B,multiset(A),aTP_Lamp_mk(multiset(A),fun(B,multiset(A)),Uu),Uua) = Uu ).

% ATP.lambda_1062
tff(fact_9245_ATP_Olambda__1063,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_ap(assn,fun(A,assn),Uu),Uua) = Uu ).

% ATP.lambda_1063
tff(fact_9246_ATP_Olambda__1064,axiom,
    ! [A: $tType,Uu: bool,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_oz(bool,fun(A,bool),Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_1064
tff(fact_9247_ATP_Olambda__1065,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_yl(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1065
tff(fact_9248_ATP_Olambda__1066,axiom,
    ! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_abc(set(D),fun(B,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1066
tff(fact_9249_ATP_Olambda__1067,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_yg(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1067
tff(fact_9250_ATP_Olambda__1068,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_yh(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1068
tff(fact_9251_ATP_Olambda__1069,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_ari(set(B),fun(C,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1069
tff(fact_9252_ATP_Olambda__1070,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_xw(set(B),fun(B,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1070
tff(fact_9253_ATP_Olambda__1071,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_xt(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1071
tff(fact_9254_ATP_Olambda__1072,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_za(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1072
tff(fact_9255_ATP_Olambda__1073,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: C] : aa(C,set(A),aTP_Lamp_arh(set(A),fun(C,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1073
tff(fact_9256_ATP_Olambda__1074,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_mi(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1074
tff(fact_9257_ATP_Olambda__1075,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_yj(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1075
tff(fact_9258_ATP_Olambda__1076,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_ws(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).

% ATP.lambda_1076
tff(fact_9259_ATP_Olambda__1077,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_xj(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_1077
tff(fact_9260_ATP_Olambda__1078,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_adg(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).

% ATP.lambda_1078
tff(fact_9261_ATP_Olambda__1079,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_bs(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).

% ATP.lambda_1079
tff(fact_9262_ATP_Olambda__1080,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_tk(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1080
tff(fact_9263_ATP_Olambda__1081,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_ve(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1081
tff(fact_9264_ATP_Olambda__1082,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aka(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1082
tff(fact_9265_ATP_Olambda__1083,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_qf(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_1083
tff(fact_9266_ATP_Olambda__1084,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ly(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1084
tff(fact_9267_ATP_Olambda__1085,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_mj(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1085
tff(fact_9268_ATP_Olambda__1086,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_acy(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1086
tff(fact_9269_ATP_Olambda__1087,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ks(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1087
tff(fact_9270_ATP_Olambda__1088,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_tz(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1088
tff(fact_9271_ATP_Olambda__1089,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_sb(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1089
tff(fact_9272_ATP_Olambda__1090,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_kt(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1090
tff(fact_9273_ATP_Olambda__1091,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_ace(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_1091
tff(fact_9274_ATP_Olambda__1092,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_acw(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_1092
tff(fact_9275_ATP_Olambda__1093,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_px(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_1093
tff(fact_9276_ATP_Olambda__1094,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_vo(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_1094
tff(fact_9277_ATP_Olambda__1095,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_wt(A,fun(B,B)),Uu),Uua) = Uua ).

% ATP.lambda_1095
tff(fact_9278_ATP_Olambda__1096,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_ub(A,fun(list(A),bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_1096
tff(fact_9279_ATP_Olambda__1097,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aqs(A,fun(A,bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_1097
tff(fact_9280_ATP_Olambda__1098,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_ua(A,fun(list(A),bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1098
tff(fact_9281_ATP_Olambda__1099,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_jx(A,fun(B,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1099
tff(fact_9282_ATP_Olambda__1100,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_em(A,fun(A,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1100
tff(fact_9283_ATP_Olambda__1101,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ez(nat,nat),Uu) = Uu ).

% ATP.lambda_1101
tff(fact_9284_ATP_Olambda__1102,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_gs(int,int),Uu) = Uu ).

% ATP.lambda_1102
tff(fact_9285_ATP_Olambda__1103,axiom,
    ! [D: $tType,Uu: fun(D,nat)] : aa(fun(D,nat),fun(D,nat),aTP_Lamp_anf(fun(D,nat),fun(D,nat)),Uu) = Uu ).

% ATP.lambda_1103
tff(fact_9286_ATP_Olambda__1104,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_aay(B,B),Uu) = Uu ).

% ATP.lambda_1104
tff(fact_9287_ATP_Olambda__1105,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aet(A,A),Uu) = Uu ) ).

% ATP.lambda_1105
tff(fact_9288_ATP_Olambda__1106,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_vd(A,A),Uu) = Uu ) ).

% ATP.lambda_1106
tff(fact_9289_ATP_Olambda__1107,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_mm(A,A),Uu) = Uu ) ).

% ATP.lambda_1107
tff(fact_9290_ATP_Olambda__1108,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_nq(A,A),Uu) = Uu ) ).

% ATP.lambda_1108
tff(fact_9291_ATP_Olambda__1109,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rm(A,A),Uu) = Uu ) ).

% ATP.lambda_1109
tff(fact_9292_ATP_Olambda__1110,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_by(A,A),Uu) = Uu ) ).

% ATP.lambda_1110
tff(fact_9293_ATP_Olambda__1111,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_cq(A,A),Uu) = Uu ).

% ATP.lambda_1111
tff(fact_9294_ATP_Olambda__1112,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_pa(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_1112
tff(fact_9295_ATP_Olambda__1113,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(bool),aTP_Lamp_art(A,set(bool)),Uu) = top_top(set(bool)) ).

% ATP.lambda_1113
tff(fact_9296_ATP_Olambda__1114,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_xu(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_1114
tff(fact_9297_ATP_Olambda__1115,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_pr(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1115
tff(fact_9298_ATP_Olambda__1116,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_nc(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_1116
tff(fact_9299_ATP_Olambda__1117,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_mo(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_1117
tff(fact_9300_ATP_Olambda__1118,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_ps(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_1118
tff(fact_9301_ATP_Olambda__1119,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_xs(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1119
tff(fact_9302_ATP_Olambda__1120,axiom,
    ! [B: $tType,Uu: B] : aa(B,nat,aTP_Lamp_alv(B,nat),Uu) = zero_zero(nat) ).

% ATP.lambda_1120
tff(fact_9303_ATP_Olambda__1121,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_do(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1121
tff(fact_9304_ATP_Olambda__1122,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_va(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1122
tff(fact_9305_ATP_Olambda__1123,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_bz(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1123
tff(fact_9306_ATP_Olambda__1124,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_amb(A,nat),Uu) = zero_zero(nat) ).

% ATP.lambda_1124
tff(fact_9307_ATP_Olambda__1125,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_bx(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1125
tff(fact_9308_ATP_Olambda__1126,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_eg(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1126
tff(fact_9309_ATP_Olambda__1127,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_la(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_1127
tff(fact_9310_ATP_Olambda__1128,axiom,
    ! [A: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_bu(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1128
tff(fact_9311_ATP_Olambda__1129,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_lx(B,option(A)),Uu) = none(A) ).

% ATP.lambda_1129
tff(fact_9312_ATP_Olambda__1130,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_qt(A,option(C)),Uu) = none(C) ).

% ATP.lambda_1130
tff(fact_9313_ATP_Olambda__1131,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_bk(A,option(B)),Uu) = none(B) ).

% ATP.lambda_1131
tff(fact_9314_ATP_Olambda__1132,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_agk(A,B),Uu) = undefined(B) ).

% ATP.lambda_1132
tff(fact_9315_ATP_Olambda__1133,axiom,
    ! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_bh(product_prod(heap_ext(product_unit),set(nat)),bool),Uu))
    <=> $false ) ).

% ATP.lambda_1133
tff(fact_9316_ATP_Olambda__1134,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_lm(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1134
tff(fact_9317_ATP_Olambda__1135,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_ae(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1135
tff(fact_9318_ATP_Olambda__1136,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ll(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1136
tff(fact_9319_ATP_Olambda__1137,axiom,
    ! [A: $tType,Uu: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aTP_Lamp_apj(fun(A,bool),bool),Uu))
    <=> $true ) ).

% ATP.lambda_1137
tff(fact_9320_ATP_Olambda__1138,axiom,
    ! [D: $tType,Uu: D] :
      ( pp(aa(D,bool,aTP_Lamp_any(D,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1138
tff(fact_9321_ATP_Olambda__1139,axiom,
    ! [C: $tType,Uu: C] :
      ( pp(aa(C,bool,aTP_Lamp_aoa(C,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1139
tff(fact_9322_ATP_Olambda__1140,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_bm(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1140
tff(fact_9323_ATP_Olambda__1141,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_sw(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1141

% Type constructors (794)
tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(heap_Time_Heap(A20)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(heap_Time_Heap(A20)) ) ).

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,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(heap_ext(A20)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(heap_ext(A20)) ) ).

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,
    ! [A20: $tType,A21: $tType] :
      ( ( typerep(A20)
        & typerep(A21) )
     => code_term_of(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( enum(A20)
        & enum(A21) )
     => enum(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( typerep(A20)
        & typerep(A21) )
     => typerep(product_prod(A20,A21)) ) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_14,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_15,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(multiset(A20)) ) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_16,axiom,
    typerep(assn) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_17,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(option(A20)) ) ).

tff(tcon_Option_Ooption___Enum_Oenum_18,axiom,
    ! [A20: $tType] :
      ( enum(A20)
     => enum(option(A20)) ) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_19,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(option(A20)) ) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_20,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(filter(A20)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_21,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(filter(A20)) ) ).

tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_22,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( typerep(A20)
        & typerep(A21) )
     => code_term_of(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___Enum_Oenum_23,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( enum(A20)
        & enum(A21) )
     => enum(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___Typerep_Otyperep_24,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( typerep(A20)
        & typerep(A21) )
     => typerep(sum_sum(A20,A21)) ) ).

tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_25,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(array(A20)) ) ).

tff(tcon_Heap_Oarray___Typerep_Otyperep_26,axiom,
    ! [A20: $tType] : typerep(array(A20)) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_27,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(list(A20)) ) ).

tff(tcon_List_Olist___Typerep_Otyperep_28,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(list(A20)) ) ).

tff(tcon_Heap_Oref___Code__Evaluation_Oterm__of_29,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(ref(A20)) ) ).

tff(tcon_Heap_Oref___Typerep_Otyperep_30,axiom,
    ! [A20: $tType] : typerep(ref(A20)) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_31,axiom,
    code_term_of(bool) ).

tff(tcon_HOL_Obool___Enum_Oenum_32,axiom,
    enum(bool) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_33,axiom,
    typerep(bool) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_34,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(set(A20)) ) ).

tff(tcon_Set_Oset___Enum_Oenum_35,axiom,
    ! [A20: $tType] :
      ( enum(A20)
     => enum(set(A20)) ) ).

tff(tcon_Set_Oset___Typerep_Otyperep_36,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(set(A20)) ) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_37,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_38,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_39,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Typerep_Otyperep_40,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_41,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_42,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_43,axiom,
    code_term_of(int) ).

tff(tcon_Int_Oint___Typerep_Otyperep_44,axiom,
    typerep(int) ).

tff(tcon_itself___Code__Evaluation_Oterm__of_45,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => code_term_of(itself(A20)) ) ).

tff(tcon_itself___Typerep_Otyperep_46,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => typerep(itself(A20)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of_47,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( typerep(A20)
        & typerep(A21) )
     => code_term_of(fun(A20,A21)) ) ).

tff(tcon_fun___Enum_Oenum_48,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( enum(A20)
        & enum(A21) )
     => enum(fun(A20,A21)) ) ).

tff(tcon_fun___Typerep_Otyperep_49,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( typerep(A20)
        & typerep(A21) )
     => typerep(fun(A20,A21)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple6319245703460814977attice(A21)
     => condit1219197933456340205attice(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple592849572758109894attice(A21)
     => comple592849572758109894attice(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple489889107523837845lgebra(A21)
     => comple489889107523837845lgebra(fun(A20,A21)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( cl_HOL_Oequal(A20)
        & quickc3360725361186068524ustive(A20)
        & quickc3360725361186068524ustive(A21) )
     => quickc3360725361186068524ustive(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounde4967611905675639751up_bot(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounde4346867609351753570nf_top(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple6319245703460814977attice(A21)
     => comple6319245703460814977attice(fun(A20,A21)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( cl_HOL_Oequal(A20)
        & quickc658316121487927005ustive(A20)
        & quickc658316121487927005ustive(A21) )
     => quickc658316121487927005ustive(fun(A20,A21)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A20: $tType,A21: $tType] :
      ( boolea8198339166811842893lgebra(A21)
     => boolea8198339166811842893lgebra(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounded_lattice_top(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounded_lattice_bot(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple6319245703460814977attice(A21)
     => comple9053668089753744459l_ccpo(fun(A20,A21)) ) ).

tff(tcon_fun___Quickcheck__Random_Orandom,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( code_term_of(A20)
        & cl_HOL_Oequal(A20)
        & quickcheck_random(A21) )
     => quickcheck_random(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_sup(A21)
     => semilattice_sup(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_inf(A21)
     => semilattice_inf(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( distrib_lattice(A21)
     => distrib_lattice(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounded_lattice(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A20: $tType,A21: $tType] :
      ( complete_Sup(A21)
     => complete_Sup(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A20: $tType,A21: $tType] :
      ( complete_Inf(A21)
     => complete_Inf(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A20: $tType,A21: $tType] :
      ( order_top(A21)
     => order_top(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A20: $tType,A21: $tType] :
      ( order_bot(A21)
     => order_bot(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A20: $tType,A21: $tType] :
      ( preorder(A21)
     => preorder(fun(A20,A21)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( finite_finite(A20)
        & finite_finite(A21) )
     => finite_finite(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( lattice(A21)
     => lattice(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A20: $tType,A21: $tType] :
      ( order(A21)
     => order(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A20: $tType,A21: $tType] :
      ( top(A21)
     => top(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ord(A21)
     => ord(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bot(A21)
     => bot(fun(A20,A21)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A20: $tType,A21: $tType] :
      ( uminus(A21)
     => uminus(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Osup,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_sup(A21)
     => sup(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Oinf,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_inf(A21)
     => inf(fun(A20,A21)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A20: $tType,A21: $tType] :
      ( minus(A21)
     => minus(fun(A20,A21)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( enum(A20)
        & cl_HOL_Oequal(A21) )
     => cl_HOL_Oequal(fun(A20,A21)) ) ).

tff(tcon_itself___Quickcheck__Random_Orandom_50,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => quickcheck_random(itself(A20)) ) ).

tff(tcon_itself___HOL_Oequal_51,axiom,
    ! [A20: $tType] : cl_HOL_Oequal(itself(A20)) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_52,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_53,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___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_54,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___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___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(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_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_55,axiom,
    quickcheck_random(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_56,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_57,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_58,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___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OSup_59,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_60,axiom,
    complete_Inf(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___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___Orderings_Opreorder_61,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_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_62,axiom,
    lattice(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_63,axiom,
    order(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_64,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_65,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Lattices_Osup_66,axiom,
    sup(int) ).

tff(tcon_Int_Oint___Lattices_Oinf_67,axiom,
    inf(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___Groups_Ominus_68,axiom,
    minus(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___Divides_Ounique__euclidean__semiring__numeral_78,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_79,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_80,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_81,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_82,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_83,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_84,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_85,axiom,
    quickc3360725361186068524ustive(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_86,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_87,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_88,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_89,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_90,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_91,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_92,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_93,axiom,
    quickc658316121487927005ustive(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_94,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_95,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_96,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_97,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_98,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_99,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_100,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_101,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_102,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_103,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Random_Orandom_104,axiom,
    quickcheck_random(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_105,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_106,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_107,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_108,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_109,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_110,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_111,axiom,
    comm_monoid_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_112,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_113,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_114,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_115,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_116,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_117,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_118,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_119,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_120,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_121,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_122,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_123,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_124,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_125,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_126,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_127,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_128,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_129,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_130,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_131,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_132,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_133,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_134,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_135,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_136,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_137,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_138,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_139,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_140,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_141,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_142,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_143,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Lattices_Osup_144,axiom,
    sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Oinf_145,axiom,
    inf(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_146,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_147,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_148,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_149,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_150,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_151,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_152,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_153,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_154,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_155,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_156,axiom,
    quickc3360725361186068524ustive(num) ).

tff(tcon_Num_Onum___Quickcheck__Random_Orandom_157,axiom,
    quickcheck_random(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_158,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_159,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_160,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_161,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_162,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_163,axiom,
    plus(num) ).

tff(tcon_Num_Onum___HOL_Oequal_164,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Num_Onum___Nat_Osize_165,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_166,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_167,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_168,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_169,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_170,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_171,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_172,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_173,axiom,
    quickc3360725361186068524ustive(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_174,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_175,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_176,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_177,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_178,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_179,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_180,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_181,axiom,
    quickc658316121487927005ustive(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_182,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_183,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_184,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_185,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_186,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_187,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_188,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_189,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_190,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_191,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_192,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_193,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_194,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_195,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_196,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Random_Orandom_197,axiom,
    quickcheck_random(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_198,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_199,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_200,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_201,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_202,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_203,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_204,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_205,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_206,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_207,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_208,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_209,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_210,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_211,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_212,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_213,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_214,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_215,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_216,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_217,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_218,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_219,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_220,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_221,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_222,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_223,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_224,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_225,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_226,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_227,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_228,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_229,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_230,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_231,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_232,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_233,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_234,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_235,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_236,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_237,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_238,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_239,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Lattices_Osup_240,axiom,
    sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Oinf_241,axiom,
    inf(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_242,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_243,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_244,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_245,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_246,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_247,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_248,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_249,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_250,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_251,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_252,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_253,axiom,
    ! [A20: $tType] : condit1219197933456340205attice(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_254,axiom,
    ! [A20: $tType] : comple592849572758109894attice(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_255,axiom,
    ! [A20: $tType] : comple489889107523837845lgebra(set(A20)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_256,axiom,
    ! [A20: $tType] :
      ( quickc3360725361186068524ustive(A20)
     => quickc3360725361186068524ustive(set(A20)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_257,axiom,
    ! [A20: $tType] : bounde4967611905675639751up_bot(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_258,axiom,
    ! [A20: $tType] : bounde4346867609351753570nf_top(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_259,axiom,
    ! [A20: $tType] : comple6319245703460814977attice(set(A20)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_260,axiom,
    ! [A20: $tType] :
      ( quickc658316121487927005ustive(A20)
     => quickc658316121487927005ustive(set(A20)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_261,axiom,
    ! [A20: $tType] : boolea8198339166811842893lgebra(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_262,axiom,
    ! [A20: $tType] : bounded_lattice_top(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_263,axiom,
    ! [A20: $tType] : bounded_lattice_bot(set(A20)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_264,axiom,
    ! [A20: $tType] : comple9053668089753744459l_ccpo(set(A20)) ).

tff(tcon_Set_Oset___Quickcheck__Random_Orandom_265,axiom,
    ! [A20: $tType] :
      ( quickcheck_random(A20)
     => quickcheck_random(set(A20)) ) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_266,axiom,
    ! [A20: $tType] : semilattice_sup(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_267,axiom,
    ! [A20: $tType] : semilattice_inf(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_268,axiom,
    ! [A20: $tType] : distrib_lattice(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_269,axiom,
    ! [A20: $tType] : bounded_lattice(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_270,axiom,
    ! [A20: $tType] : complete_Sup(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_271,axiom,
    ! [A20: $tType] : complete_Inf(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_272,axiom,
    ! [A20: $tType] : order_top(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_273,axiom,
    ! [A20: $tType] : order_bot(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_274,axiom,
    ! [A20: $tType] : preorder(set(A20)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_275,axiom,
    ! [A20: $tType] :
      ( finite_finite(A20)
     => finite_finite(set(A20)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_276,axiom,
    ! [A20: $tType] : lattice(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oorder_277,axiom,
    ! [A20: $tType] : order(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Otop_278,axiom,
    ! [A20: $tType] : top(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oord_279,axiom,
    ! [A20: $tType] : ord(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Obot_280,axiom,
    ! [A20: $tType] : bot(set(A20)) ).

tff(tcon_Set_Oset___Groups_Ouminus_281,axiom,
    ! [A20: $tType] : uminus(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Osup_282,axiom,
    ! [A20: $tType] : sup(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Oinf_283,axiom,
    ! [A20: $tType] : inf(set(A20)) ).

tff(tcon_Set_Oset___Groups_Ominus_284,axiom,
    ! [A20: $tType] : minus(set(A20)) ).

tff(tcon_Set_Oset___HOL_Oequal_285,axiom,
    ! [A20: $tType] :
      ( cl_HOL_Oequal(A20)
     => cl_HOL_Oequal(set(A20)) ) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_286,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_287,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_288,axiom,
    comple489889107523837845lgebra(bool) ).

tff(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_289,axiom,
    quickc3360725361186068524ustive(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_290,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_291,axiom,
    bounde4346867609351753570nf_top(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_292,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_293,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_294,axiom,
    bounded_lattice_top(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_295,axiom,
    bounded_lattice_bot(bool) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_296,axiom,
    comple9053668089753744459l_ccpo(bool) ).

tff(tcon_HOL_Obool___Quickcheck__Random_Orandom_297,axiom,
    quickcheck_random(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_298,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_299,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_300,axiom,
    distrib_lattice(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_301,axiom,
    bounded_lattice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_302,axiom,
    complete_Sup(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_303,axiom,
    complete_Inf(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_304,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_305,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_306,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_307,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_308,axiom,
    finite_finite(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_309,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_310,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Otop_311,axiom,
    top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_312,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Orderings_Obot_313,axiom,
    bot(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_314,axiom,
    uminus(bool) ).

tff(tcon_HOL_Obool___Lattices_Osup_315,axiom,
    sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Oinf_316,axiom,
    inf(bool) ).

tff(tcon_HOL_Obool___Groups_Ominus_317,axiom,
    minus(bool) ).

tff(tcon_HOL_Obool___Heap_Oheap_318,axiom,
    heap(bool) ).

tff(tcon_HOL_Obool___HOL_Oequal_319,axiom,
    cl_HOL_Oequal(bool) ).

tff(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_320,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => quickc3360725361186068524ustive(ref(A20)) ) ).

tff(tcon_Heap_Oref___Quickcheck__Random_Orandom_321,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => quickcheck_random(ref(A20)) ) ).

tff(tcon_Heap_Oref___Heap_Oheap_322,axiom,
    ! [A20: $tType] : heap(ref(A20)) ).

tff(tcon_Heap_Oref___HOL_Oequal_323,axiom,
    ! [A20: $tType] : cl_HOL_Oequal(ref(A20)) ).

tff(tcon_Heap_Oref___Nat_Osize_324,axiom,
    ! [A20: $tType] : size(ref(A20)) ).

tff(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_325,axiom,
    ! [A20: $tType] :
      ( quickc3360725361186068524ustive(A20)
     => quickc3360725361186068524ustive(list(A20)) ) ).

tff(tcon_List_Olist___Quickcheck__Random_Orandom_326,axiom,
    ! [A20: $tType] :
      ( quickcheck_random(A20)
     => quickcheck_random(list(A20)) ) ).

tff(tcon_List_Olist___Heap_Oheap_327,axiom,
    ! [A20: $tType] :
      ( heap(A20)
     => heap(list(A20)) ) ).

tff(tcon_List_Olist___HOL_Oequal_328,axiom,
    ! [A20: $tType] : cl_HOL_Oequal(list(A20)) ).

tff(tcon_List_Olist___Nat_Osize_329,axiom,
    ! [A20: $tType] : size(list(A20)) ).

tff(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_330,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => quickc3360725361186068524ustive(array(A20)) ) ).

tff(tcon_Heap_Oarray___Quickcheck__Random_Orandom_331,axiom,
    ! [A20: $tType] :
      ( typerep(A20)
     => quickcheck_random(array(A20)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_332,axiom,
    ! [A20: $tType] : heap(array(A20)) ).

tff(tcon_Heap_Oarray___HOL_Oequal_333,axiom,
    ! [A20: $tType] : cl_HOL_Oequal(array(A20)) ).

tff(tcon_Heap_Oarray___Nat_Osize_334,axiom,
    ! [A20: $tType] : size(array(A20)) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_335,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( quickc3360725361186068524ustive(A20)
        & quickc3360725361186068524ustive(A21) )
     => quickc3360725361186068524ustive(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_336,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( quickcheck_random(A20)
        & quickcheck_random(A21) )
     => quickcheck_random(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_337,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( finite_finite(A20)
        & finite_finite(A21) )
     => finite_finite(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___Heap_Oheap_338,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( heap(A20)
        & heap(A21) )
     => heap(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___HOL_Oequal_339,axiom,
    ! [A20: $tType,A21: $tType] : cl_HOL_Oequal(sum_sum(A20,A21)) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_340,axiom,
    ! [A20: $tType,A21: $tType] : size(sum_sum(A20,A21)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_341,axiom,
    ! [A20: $tType] : condit1219197933456340205attice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_342,axiom,
    ! [A20: $tType] : bounde4967611905675639751up_bot(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_343,axiom,
    ! [A20: $tType] : bounde4346867609351753570nf_top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_344,axiom,
    ! [A20: $tType] : comple6319245703460814977attice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_345,axiom,
    ! [A20: $tType] : bounded_lattice_top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_346,axiom,
    ! [A20: $tType] : bounded_lattice_bot(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_347,axiom,
    ! [A20: $tType] : comple9053668089753744459l_ccpo(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_348,axiom,
    ! [A20: $tType] : semilattice_sup(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_349,axiom,
    ! [A20: $tType] : semilattice_inf(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_350,axiom,
    ! [A20: $tType] : distrib_lattice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_351,axiom,
    ! [A20: $tType] : bounded_lattice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_352,axiom,
    ! [A20: $tType] : complete_Sup(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_353,axiom,
    ! [A20: $tType] : complete_Inf(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_354,axiom,
    ! [A20: $tType] : order_top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_355,axiom,
    ! [A20: $tType] : order_bot(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_356,axiom,
    ! [A20: $tType] : preorder(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_357,axiom,
    ! [A20: $tType] : lattice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_358,axiom,
    ! [A20: $tType] : order(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_359,axiom,
    ! [A20: $tType] : top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_360,axiom,
    ! [A20: $tType] : ord(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_361,axiom,
    ! [A20: $tType] : bot(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Osup_362,axiom,
    ! [A20: $tType] : sup(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Oinf_363,axiom,
    ! [A20: $tType] : inf(filter(A20)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_364,axiom,
    ! [A20: $tType] :
      ( cl_HOL_Oequal(A20)
     => cl_HOL_Oequal(filter(A20)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_365,axiom,
    ! [A20: $tType] :
      ( comple5582772986160207858norder(A20)
     => condit6923001295902523014norder(option(A20)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_366,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => condit1219197933456340205attice(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_367,axiom,
    ! [A20: $tType] :
      ( comple592849572758109894attice(A20)
     => comple592849572758109894attice(option(A20)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_368,axiom,
    ! [A20: $tType] :
      ( quickc3360725361186068524ustive(A20)
     => quickc3360725361186068524ustive(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_369,axiom,
    ! [A20: $tType] :
      ( lattice(A20)
     => bounde4967611905675639751up_bot(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_370,axiom,
    ! [A20: $tType] :
      ( bounded_lattice_top(A20)
     => bounde4346867609351753570nf_top(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
    ! [A20: $tType] :
      ( comple5582772986160207858norder(A20)
     => comple5582772986160207858norder(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_371,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple6319245703460814977attice(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_372,axiom,
    ! [A20: $tType] :
      ( bounded_lattice_top(A20)
     => bounded_lattice_top(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_373,axiom,
    ! [A20: $tType] :
      ( lattice(A20)
     => bounded_lattice_bot(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_374,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple9053668089753744459l_ccpo(option(A20)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Random_Orandom_375,axiom,
    ! [A20: $tType] :
      ( quickcheck_random(A20)
     => quickcheck_random(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_376,axiom,
    ! [A20: $tType] :
      ( semilattice_sup(A20)
     => semilattice_sup(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_377,axiom,
    ! [A20: $tType] :
      ( semilattice_inf(A20)
     => semilattice_inf(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_378,axiom,
    ! [A20: $tType] :
      ( distrib_lattice(A20)
     => distrib_lattice(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_379,axiom,
    ! [A20: $tType] :
      ( bounded_lattice_top(A20)
     => bounded_lattice(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OSup_380,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => complete_Sup(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OInf_381,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => complete_Inf(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_382,axiom,
    ! [A20: $tType] :
      ( wellorder(A20)
     => wellorder(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_383,axiom,
    ! [A20: $tType] :
      ( order_top(A20)
     => order_top(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_384,axiom,
    ! [A20: $tType] :
      ( order(A20)
     => order_bot(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_385,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => preorder(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_386,axiom,
    ! [A20: $tType] :
      ( linorder(A20)
     => linorder(option(A20)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_387,axiom,
    ! [A20: $tType] :
      ( finite_finite(A20)
     => finite_finite(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_388,axiom,
    ! [A20: $tType] :
      ( lattice(A20)
     => lattice(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_389,axiom,
    ! [A20: $tType] :
      ( order(A20)
     => order(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_390,axiom,
    ! [A20: $tType] :
      ( order_top(A20)
     => top(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_391,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => ord(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_392,axiom,
    ! [A20: $tType] :
      ( order(A20)
     => bot(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Osup_393,axiom,
    ! [A20: $tType] :
      ( sup(A20)
     => sup(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Oinf_394,axiom,
    ! [A20: $tType] :
      ( inf(A20)
     => inf(option(A20)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_395,axiom,
    ! [A20: $tType] :
      ( heap(A20)
     => heap(option(A20)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_396,axiom,
    ! [A20: $tType] : cl_HOL_Oequal(option(A20)) ).

tff(tcon_Option_Ooption___Nat_Osize_397,axiom,
    ! [A20: $tType] : size(option(A20)) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_398,axiom,
    bounde4967611905675639751up_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_399,axiom,
    bounde4346867609351753570nf_top(assn) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_400,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_401,axiom,
    bounded_lattice_top(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_402,axiom,
    bounded_lattice_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_403,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_404,axiom,
    semilattice_inf(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_405,axiom,
    distrib_lattice(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_406,axiom,
    bounded_lattice(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_407,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_408,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_409,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_410,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_411,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_412,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_413,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_414,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_415,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Otop_416,axiom,
    top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_417,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_418,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_419,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osup_420,axiom,
    sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Oinf_421,axiom,
    inf(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_422,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_423,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_424,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_425,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_426,axiom,
    dvd(assn) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_427,axiom,
    ! [A20: $tType] :
      ( quickc3360725361186068524ustive(A20)
     => quickc3360725361186068524ustive(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_428,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => ordere6658533253407199908up_add(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_429,axiom,
    ! [A20: $tType] : cancel_semigroup_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_430,axiom,
    ! [A20: $tType] :
      ( quickcheck_random(A20)
     => quickcheck_random(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_431,axiom,
    ! [A20: $tType] : comm_monoid_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OSup_432,axiom,
    ! [A20: $tType] : complete_Sup(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OInf_433,axiom,
    ! [A20: $tType] : complete_Inf(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_434,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => preorder(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_435,axiom,
    ! [A20: $tType] : monoid_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_436,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => order(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_437,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => ord(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_438,axiom,
    ! [A20: $tType] : minus(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_439,axiom,
    ! [A20: $tType] : zero(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_440,axiom,
    ! [A20: $tType] : plus(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___HOL_Oequal_441,axiom,
    ! [A20: $tType] :
      ( cl_HOL_Oequal(A20)
     => cl_HOL_Oequal(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_442,axiom,
    ! [A20: $tType] : size(multiset(A20)) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_443,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( quickc3360725361186068524ustive(A20)
        & quickc3360725361186068524ustive(A21) )
     => quickc3360725361186068524ustive(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_444,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( quickc658316121487927005ustive(A20)
        & quickc658316121487927005ustive(A21) )
     => quickc658316121487927005ustive(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_445,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( quickcheck_random(A20)
        & quickcheck_random(A21) )
     => quickcheck_random(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_446,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( finite_finite(A20)
        & finite_finite(A21) )
     => finite_finite(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_447,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( heap(A20)
        & heap(A21) )
     => heap(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_448,axiom,
    ! [A20: $tType,A21: $tType] : cl_HOL_Oequal(product_prod(A20,A21)) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_449,axiom,
    ! [A20: $tType,A21: $tType] : size(product_prod(A20,A21)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_450,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_451,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_452,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_453,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_454,axiom,
    quickc3360725361186068524ustive(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_455,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_456,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_457,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_458,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_459,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_460,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_461,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_462,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_463,axiom,
    quickcheck_random(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_464,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_465,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_466,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_467,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_468,axiom,
    complete_Sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_469,axiom,
    complete_Inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_470,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_471,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_472,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_473,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_474,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_475,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_476,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_477,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_478,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_479,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_480,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_481,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osup_482,axiom,
    sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Oinf_483,axiom,
    inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_484,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_485,axiom,
    heap(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_486,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_487,axiom,
    ! [A20: $tType] :
      ( quickcheck_random(A20)
     => quickcheck_random(heap_ext(A20)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_488,axiom,
    ! [A20: $tType] : cl_HOL_Oequal(heap_ext(A20)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_489,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_490,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_491,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_492,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_493,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_494,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_495,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_496,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_497,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_498,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_499,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_500,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_501,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_502,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_503,axiom,
    quickc3360725361186068524ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_504,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_505,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_506,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_507,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_508,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_509,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_510,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_511,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_512,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_513,axiom,
    quickc658316121487927005ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_514,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_515,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_516,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_517,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_518,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_519,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_520,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_521,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_522,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_523,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_524,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_525,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_526,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_527,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_528,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_529,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_530,axiom,
    quickcheck_random(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_531,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_532,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_533,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_534,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_535,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_536,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_537,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_538,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_539,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_540,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_541,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_542,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_543,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_544,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_545,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_546,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_547,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_548,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_549,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_550,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_551,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_552,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_553,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_554,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_555,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_556,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_557,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_558,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_559,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_560,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_561,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_562,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_563,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_564,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_565,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_566,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_567,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_568,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_569,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_570,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_571,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_572,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_573,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_574,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_575,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_576,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_577,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_578,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_579,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_580,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_581,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_582,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_583,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_584,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_585,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_586,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_587,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_588,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_589,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_590,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_591,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_592,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_593,axiom,
    quickc3360725361186068524ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_594,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_595,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_596,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_597,axiom,
    linord4140545234300271783up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_598,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_599,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_600,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_601,axiom,
    quickc658316121487927005ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_602,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_603,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_604,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_605,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_606,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_607,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_608,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_609,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_610,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_611,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_612,axiom,
    quickcheck_random(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_613,axiom,
    ab_semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_614,axiom,
    semiring_1_cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_615,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_616,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_617,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_618,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_619,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_620,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_621,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_622,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_623,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_624,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_625,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_626,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_627,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_628,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_629,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_630,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_631,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_632,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_633,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_634,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_635,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_636,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_637,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_638,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_639,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_640,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Otimes_641,axiom,
    times(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ominus_642,axiom,
    minus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_643,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_644,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_645,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oplus_646,axiom,
    plus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_647,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_648,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___HOL_Oequal_649,axiom,
    cl_HOL_Oequal(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_650,axiom,
    size(code_natural) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_651,axiom,
    ! [A20: $tType] :
      ( quickcheck_random(A20)
     => quickcheck_random(heap_Time_Heap(A20)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_652,axiom,
    ! [A20: $tType] : cl_HOL_Oequal(heap_Time_Heap(A20)) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_653,axiom,
    ! [A20: $tType] : size(heap_Time_Heap(A20)) ).

% Helper facts (16)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(aa(fun(A,bool),bool,fAll(A),P))
      | pp(aa(A,bool,P,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa(bool,bool,fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa(bool,bool,fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(fconj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fdisj(P,Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(fdisj(P,Q)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(a,assn,q,r2)),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)),h),hoare_new_addrs(h2,as,h)))) ).

%------------------------------------------------------------------------------